Rearranging Formulas to Highlight Quantities - Algebra 2

In physics, kinetic energy is $KE=\tfrac{1}{2}mv^2$ , where $m$ is mass and $v$ is speed. Solve $KE=\tfrac{1}{2}mv^2$ for $v$ (include $\pm$ where appropriate).
Answer: $v=\pm\sqrt{\dfrac{2KE}{m}}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. Starting with KE = (1/2)mv², we need to isolate v. First multiply both sides by 2: 2KE = mv². Then divide both sides by m: 2KE/m = v². Finally, take the square root: v = ±√(2KE/m). The ± is crucial because squaring either positive or negative v gives the same v². Choice B correctly isolates v through multiplication, division, and taking the square root to get v = ±√(2KE/m). Choice A forgets the square root, Choice C has the wrong coefficient (should be 2KE not KE), and Choice D incorrectly inverts the expression under the square root. Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in xy + xz = w), factor it out first: x(y + z) = w, then x = w/(y + z). If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember ± if the formula context allows both positive and negative (though often context restricts to positive only, like radius r ≥ 0). Physics and geometry formulas usually want positive values only!
In physics, the kinematic equation $v^2 = u^2 + 2as$ relates final speed $v$ , initial speed $u$ , acceleration $a$ , and displacement $s$ . Solve $v^2 = u^2 + 2as$ for $a$ (treat the other variables as constants, using the same inverse-operation reasoning as solving a numeric equation).
Answer: $a=\dfrac{v^2-u^2}{2s}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving V = πr²h for r is like solving 100 = 3.14·r²·5 for r: divide by π and h, then take square root. Starting with v² = u² + 2as, we need to isolate a. First subtract u² from both sides: v² - u² = 2as. Then divide both sides by 2s: (v² - u²)/(2s) = a, which gives us a = (v² - u²)/(2s). Choice B correctly isolates a through subtraction and division to get a = (v² - u²)/(2s). Choice A forgets to divide by 2, Choice C incorrectly switches the order in the numerator (should be v² - u², not u² - v²), and Choice D inverts the entire fraction. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc.
In finance, compound growth can be modeled by $A=P(1+r)^t$ , where $A$ is the final amount, $P$ is the initial principal, $r$ is the rate per period, and $t$ is the number of periods. Solve $A=P(1+r)^t$ for $P$ .
Answer: $P=\dfrac{A}{(1+r)^t}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving V = πr²h for r is like solving 100 = 3.14·r²·5 for r: divide by π and h, then take square root. Starting with A = $P(1+r)^t$ , we need to isolate P. Since P is multiplied by $(1+r)^t$ , we divide both sides by $(1+r)^t$ : A/ $(1+r)^t$ = P, which gives us P = A/ $(1+r)^t$ . Choice B correctly isolates P through division to get P = A/ $(1+r)^t$ . Choice A incorrectly multiplies instead of dividing, Choice C inverts the entire fraction, and Choice D incorrectly writes the exponent as $r^t$ instead of $(1+r)^t$ . The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc.
In physics, the kinematic equation $v^2 = u^2 + 2as$ relates final speed $v$ , initial speed $u$ , acceleration $a$ , and displacement $s$ . Solve $v^2 = u^2 + 2as$ for $a$ (treating the other variables as constants, as in solving a numeric equation).
Answer: $a = \dfrac{v^2-u^2}{2s}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. Starting with $v^2 = u^2 + 2as$ , subtract $u^2$ from both sides to get $v^2 - u^2 = 2as$ , then divide both sides by $2s$ to isolate a, resulting in $a = \dfrac{v^2 - u^2}{2s}$ . Choice B correctly isolates a through subtraction and division to get $a = \dfrac{v^2 - u^2}{2s}$ . A common distractor like choice C might forget to switch the signs when subtracting $u^2$ , but remember that $v^2 - u^2$ is positive if v > u, so the order matters for the physics context. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. Keep practicing these, and you'll master kinematic equations in no time!
In chemistry, the ideal gas law is $PV=nRT$ , where $P$ is pressure, $V$ is volume, $n$ is moles, $R$ is a constant, and $T$ is temperature. Solve for $T$ in terms of $P$ , $V$ , $n$ , and $R$ .
Answer: $T=\dfrac{PV}{nR}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving V = πr²h for r is like solving 100 = 3.14·r²·5 for r: divide by π and h, then take square root. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. To solve PV = nRT for T, divide both sides by nR: T = PV/(nR). Choice B correctly isolates T through division by nR to get T = PV/(nR). A common distractor like choice A fails by inverting the fraction incorrectly, which would not isolate T properly; ensure you divide by the coefficients multiplying T. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. Example: T = 2π√(L/g) solve for L: square both sides → T² = 4π²L/g, multiply by g → gT² = 4π²L, divide by 4π² → L = gT²/(4π²). Each step undoes an operation! Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in xy + xz = w), factor it out first: x(y + z) = w, then x = w/(y + z). If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember ± if the formula context allows both positive and negative (though often context restricts to positive only, like radius r ≥ 0). Physics and geometry formulas usually want positive values only!
In geometry, the area of a triangle can be written as $A = \dfrac{1}{2}bh$ , where $b$ is base and $h$ is height. Solve $A = \dfrac{1}{2}bh$ for $h$ .
Answer: $h = \dfrac{2A}{b}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. Starting with A = (1/2)bh, multiply both sides by 2 to get 2A = bh, then divide by b to isolate h as h = 2A/b. Choice C correctly isolates h through multiplication and division to get h = 2A/b. A distractor like choice A might swap the 2 in the denominator, but multiplying by 2 clears the fraction properly. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. Great foundation for geometry—keep building!
A right circular cone has volume $V=\dfrac{1}{3}\pi r^2 h$ , where $r$ is the radius and $h$ is the height. Solve for $r$ in terms of $V$ and $h$ .
Answer: $r=\sqrt{\dfrac{3V}{\pi h}}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving V = πr²h for r is like solving 100 = 3.14·r²·5 for r: divide by π and h, then take square root. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. To solve V = (1/3)πr²h for r, first multiply both sides by 3/πh to isolate r²: (3V)/(πh) = r². Then take the square root: r = √((3V)/(πh)). Choice B correctly isolates r through multiplying by the reciprocal and taking the square root to get r = √(3V/(πh)). A common distractor like choice A fails by omitting the square root, leaving r² unaddressed; always apply the inverse of the power. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. Example: T = 2π√(L/g) solve for L: square both sides → T² = 4π²L/g, multiply by g → gT² = 4π²L, divide by 4π² → L = gT²/(4π²). Each step undoes an operation! Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in xy + xz = w), factor it out first: x(y + z) = w, then x = w/(y + z). If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember ± if the formula context allows both positive and negative (though often context restricts to positive only, like radius r ≥ 0). Physics and geometry formulas usually want positive values only!
In geometry, the volume of a cone is $V=\dfrac{1}{3}\pi r^2h$ , where $r$ is radius and $h$ is height. Solve $V=\dfrac{1}{3}\pi r^2h$ for $r$ .
Answer: $r=\sqrt{\dfrac{3V}{\pi h}}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. Starting with V = (1/3)πr²h, we need to isolate r. First multiply both sides by 3: 3V = πr²h. Then divide both sides by πh: 3V/(πh) = r². Finally, take the square root: r = √[3V/(πh)]. Choice B correctly isolates r through multiplication, division, and taking the square root to get r = √(3V/(πh)). Choice A forgets the factor of 3, Choice C forgets to take the square root (leaving r²), and Choice D incorrectly inverts the expression under the square root. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc.
In geometry, the area of a circle is $A=\pi r^2$ , where $r$ is the radius. Rearrange $A=\pi r^2$ to make $r$ the subject (use inverse operations as with solving an equation).
Answer: $r=\sqrt{\dfrac{A}{\pi}}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. Starting with A = πr², we need to isolate r. First divide both sides by π: A/π = r². Then take the square root of both sides: √(A/π) = r, which gives us r = √(A/π). Choice A correctly isolates r through division by π and taking the square root to get r = √(A/π). Choice B incorrectly divides by π² instead of π, Choice C forgets to take the square root (leaving r²), and Choice D incorrectly multiplies inside the square root instead of dividing. Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in xy + xz = w), factor it out first: x(y + z) = w, then x = w/(y + z). If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember ± if the formula context allows both positive and negative (though often context restricts to positive only, like radius r ≥ 0). Physics and geometry formulas usually want positive values only!
In geometry, the volume of a sphere is $V=\dfrac{4}{3}\pi r^3$ . Solve for the radius $r$ in terms of $V$ .
Answer: $r=\sqrt[3]{\dfrac{3V}{4\pi}}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving V = πr²h for r is like solving 100 = 3.14·r²·5 for r: divide by π and h, then take square root. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. To solve V = (4/3)πr³ for r, first multiply both sides by 3/4 to isolate the r³ term: (3V)/(4π) = r³. Then take the cube root of both sides: r = ∛((3V)/(4π)). Choice A correctly isolates r through multiplying by the reciprocal and taking the cube root to get r = ∛(3V/(4π)). A common distractor like choice B fails by using a square root instead of a cube root, which would be incorrect because r is cubed, not squared; always match the root to the power. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. Example: T = 2π√(L/g) solve for L: square both sides → T² = 4π²L/g, multiply by g → gT² = 4π²L, divide by 4π² → L = gT²/(4π²). Each step undoes an operation! Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in xy + xz = w), factor it out first: x(y + z) = w, then x = w/(y + z). If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember ± if the formula context allows both positive and negative (though often context restricts to positive only, like radius r ≥ 0). Physics and geometry formulas usually want positive values only!
A projectile's vertical displacement can be modeled by the kinematic equation $s=ut+\tfrac{1}{2}at^2$ , where $s$ is displacement, $u$ is initial velocity, $a$ is acceleration, and $t$ is time. Solve $s=ut+\tfrac{1}{2}at^2$ for $t$ (a quadratic in $t$ ).
Answer: $t=\dfrac{-u\pm\sqrt{u^2+2as}}{a}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. More complex rearrangements may require advanced techniques: if your target variable is squared (like $r^2$ in $A = \pi r^2$ ), you'll need square roots ( $r = \sqrt{\frac{A}{\pi}}$ ). If it appears in a denominator (like $f$ in $\frac{1}{f} = \frac{1}{a} + \frac{1}{b}$ ), you'll need to clear fractions first. If it appears with different powers (like $t$ in $s = ut + \frac{1}{2}at^2$ ), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. Starting with $s = ut + \frac{1}{2}at^2$ , we need to solve for $t$ . First multiply through by 2: $2s = 2ut + at^2$ . Rearranging: $at^2 + 2ut - 2s = 0$ . This is a quadratic in standard form with $A = a$ , $B = 2u$ , $C = -2s$ . Using the quadratic formula: $t = \frac{-2u \pm \sqrt{4u^2 + 8as}}{2a} = \frac{-2u \pm 2\sqrt{u^2 + 2as}}{2a} = \frac{-u \pm \sqrt{u^2 + 2as}}{a}$ . Choice A correctly applies the quadratic formula to get $t = \frac{-u \pm \sqrt{u^2 + 2as}}{a}$ . Choice B has the wrong sign on u, Choice C has the wrong sign under the square root (should be +2as not -2as), and Choice D incorrectly divides by 2a instead of a. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc.
A projectile's height might be modeled by a quadratic literal equation $h = ut - \dfrac{1}{2}gt^2$ , where $h$ is height, $u$ is initial vertical velocity, $g$ is gravitational acceleration, and $t$ is time. Solve $h = ut - \dfrac{1}{2}gt^2$ for $t$ .
Answer: $t = \dfrac{u \pm \sqrt{u^2 - 2gh}}{g}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. More complex rearrangements may require advanced techniques: if your target variable appears with different powers (like t in h = ut - (1/2)gt²), you may need the quadratic formula! Rewrite as (1/2)gt² - ut + h = 0, then apply quadratic formula t = [u ± √(u² - 2gh)] / g, adjusting coefficients carefully. Choice A correctly isolates t using the quadratic formula to get t = (u ± √(u² - 2gh)) / g. A distractor like choice B might flip the sign on u, but in the standard arrangement, it's positive u for the linear term. Watch for variables appearing multiple times: if your target variable appears in multiple places (like t here), rearrange to standard quadratic form first. You're handling projectiles like a pro—keep practicing quadratics!
A physics student needs to determine the acceleration of an object given the formula $F = ma + kv^2$ , where $F$ is the total force, $m$ is mass, $a$ is acceleration, $k$ is a drag coefficient, and $v$ is velocity. Which expression correctly isolates the acceleration?
Answer: $a = \frac{F - kv^2}{m}$ To isolate $a$ , subtract $kv^2$ from both sides: $F - kv^2 = ma$ , then divide by $m$ : $a = \frac{F - kv^2}{m}$ . Choice B incorrectly adds instead of subtracts $kv^2$ . Choice C fails to divide by $m$ . Choice D incorrectly distributes the division.
In chemistry, a simplified relationship can be written as $P=\dfrac{nRT}{V}$ , where $P$ is pressure, $n$ is moles, $R$ is a constant, $T$ is temperature, and $V$ is volume. Solve $P=\dfrac{nRT}{V}$ for $T$ .
Answer: $T=\dfrac{PV}{nR}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving $V = \pi r^2 h$ for r is like solving $100 = 3.14 \cdot r^2 \cdot 5$ for r: divide by $\pi$ and $h$ , then take square root. Starting with $P = \dfrac{nRT}{V}$ , we need to isolate T. First multiply both sides by V: $PV = nRT$ . Then divide both sides by nR: $\dfrac{PV}{nR} = T$ , which gives us $T = \dfrac{PV}{nR}$ . Choice A correctly isolates T through multiplication and division to get $T = \dfrac{PV}{nR}$ . Choice B inverts the expression, Choice C has only P in the numerator instead of PV, and Choice D incorrectly has nRP in the numerator. Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in $xy + xz = w$ ), factor it out first: $x(y + z) = w$ , then $x = \dfrac{w}{y + z}$ . If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember $\pm$ if the formula context allows both positive and negative (though often context restricts to positive only, like radius $r \geq 0$ ). Physics and geometry formulas usually want positive values only!
In physics, the kinematic relation $v^2=u^2+2as$ connects final speed $v$ , initial speed $u$ , acceleration $a$ , and displacement $s$ . Solve for $a$ in terms of $v$ , $u$ , and $s$ .
Answer: $a=\dfrac{v^2-u^2}{2s}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving V = πr²h for r is like solving 100 = 3.14·r²·5 for r: divide by π and h, then take square root. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. To solve v² = u² + 2as for a, first subtract u²: v² - u² = 2as. Then divide by 2s: a = (v² - u²)/(2s). Choice A correctly isolates a through subtraction and division to get a = (v² - u²)/(2s). A common distractor like choice B fails by reversing the signs in the numerator, which would give negative acceleration incorrectly; preserve the order of subtraction. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. Example: T = 2π√(L/g) solve for L: square both sides → T² = 4π²L/g, multiply by g → gT² = 4π²L, divide by 4π² → L = gT²/(4π²). Each step undoes an operation! Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in xy + xz = w), factor it out first: x(y + z) = w, then x = w/(y + z). If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember ± if the formula context allows both positive and negative (though often context restricts to positive only, like radius r ≥ 0). Physics and geometry formulas usually want positive values only!
In kinematics, the displacement of an object under constant acceleration is modeled by $s = ut + \tfrac{1}{2}at^2$ , where $s$ is displacement, $u$ is initial velocity, $a$ is acceleration, and $t$ is time. Solve for $t$ in terms of $s$ , $u$ , and $a$ .
Answer: $t=\dfrac{-u\pm\sqrt{u^2+2as}}{a}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving V = πr²h for r is like solving 100 = 3.14·r²·5 for r: divide by π and h, then take square root. More complex rearrangements may require advanced techniques: if your target variable is squared (like r² in A = πr²), you'll need square roots (r = √(A/π)). If it appears in a denominator (like f in 1/f = 1/a + 1/b), you'll need to clear fractions first. If it appears with different powers (like t in s = ut + (1/2)at²), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. To solve s = ut + (1/2)at² for t, first multiply both sides by 2 to clear the fraction: 2s = 2ut + at². Then rearrange into standard quadratic form: at² + 2ut - 2s = 0. Apply the quadratic formula t = [-2u ± √((2u)² - 4(a)(-2s))] / (2a), which simplifies to t = [-u ± √(u² + 2as)] / a, remembering to divide numerator and denominator appropriately. Choice A correctly isolates t through rearranging to quadratic form and applying the formula to get t = (-u ± √(u² + 2as)) / a. A common distractor like choice B fails by using a minus sign inside the square root, which would be incorrect because the discriminant is u² + 2as, not u² - 2as; always double-check the signs when forming the quadratic. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. Example: T = 2π√(L/g) solve for L: square both sides → T² = 4π²L/g, multiply by g → gT² = 4π²L, divide by 4π² → L = gT²/(4π²). Each step undoes an operation! Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in xy + xz = w), factor it out first: x(y + z) = w, then x = w/(y + z). If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember ± if the formula context allows both positive and negative (though often context restricts to positive only, like radius r ≥ 0). Physics and geometry formulas usually want positive values only!
The formula for the period of a pendulum is $T = 2\pi\sqrt{\frac{L}{g}}$ , where $T$ is period, $L$ is length, and $g$ is gravitational acceleration. An engineer needs to determine what length produces a specific period. Which expression correctly gives $L$ ?
Answer: $L = \frac{gT^2}{4\pi^2}$ When you encounter formula manipulation problems, you're essentially working backwards from a given equation to isolate a specific variable. This requires systematic algebraic steps to "undo" the operations around your target variable. Starting with $T = 2\pi\sqrt{\frac{L}{g}}$ , you need to isolate $L$ . First, eliminate the coefficient by dividing both sides by $2\pi$ : $\frac{T}{2\pi} = \sqrt{\frac{L}{g}}$ . Next, square both sides to eliminate the square root: $\left(\frac{T}{2\pi}\right)^2 = \frac{L}{g}$ . This gives you $\frac{T^2}{4\pi^2} = \frac{L}{g}$ . Finally, multiply both sides by $g$ to isolate $L$ : $L = \frac{gT^2}{4\pi^2}$ , which is answer choice C. Looking at the incorrect options: Choice A has $4\pi^2$ in the numerator instead of the denominator, suggesting someone inverted the fraction incorrectly. Choice B is missing the squared term on $T$ , indicating they forgot to square both sides when eliminating the square root. Choice D has $2\pi$ in the denominator instead of $4\pi^2$ , showing they didn't properly square the $2\pi$ term. Remember that when solving literal equations, work step-by-step and be especially careful when squaring expressions with coefficients— $(2\pi)^2 = 4\pi^2$ , not $2\pi^2$ . Always verify your final answer by substituting it back into the original equation to ensure it makes sense.
In algebraic modeling, suppose a quantity is defined by $y=\dfrac{x-a}{b}+c$ , where $a$ , $b$ , and $c$ are constants and $b\ne 0$ . Isolate $x$ in $y=\dfrac{x-a}{b}+c$ .
Answer: $x=b(y-c)+a$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving $V = \pi r^2 h$ for r is like solving $100 = 3.14 \cdot r^2 \cdot 5$ for r: divide by $\pi$ and h, then take square root. Starting with $y = \dfrac{x-a}{b} + c$ , we need to isolate x. First subtract c from both sides: $y - c = \dfrac{x-a}{b}$ . Then multiply both sides by b: $b(y - c) = x - a$ . Finally, add a to both sides: $b(y - c) + a = x$ , which gives us $x = b(y - c) + a$ . Choice B correctly isolates x through subtraction, multiplication, and addition to get $x = b(y - c) + a$ . Choice A incorrectly expands to by - a + c, Choice C divides instead of multiplying by b, and Choice D has the wrong sign on c inside the parentheses. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc.
A formula used in some models of electrical power is $P=\dfrac{V^2}{R}$ , where $P$ is power, $V$ is voltage, and $R$ is resistance. Solve for $V$ in terms of $P$ and $R$ .
Answer: $V=\pm\sqrt{PR}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. The same algebraic moves apply—just keep everything symbolic! For example, solving $V = \pi r^2 h$ for r is like solving $100 = 3.14 \cdot r^2 \cdot 5$ for r: divide by $\pi$ and h, then take square root. More complex rearrangements may require advanced techniques: if your target variable is squared (like $r^2$ in $A = \pi r^2$ ), you'll need square roots ( $r = \sqrt{\dfrac{A}{\pi}}$ ). If it appears in a denominator (like f in $\dfrac{1}{f} = \dfrac{1}{a} + \dfrac{1}{b}$ ), you'll need to clear fractions first. If it appears with different powers (like t in $s = ut + \dfrac{1}{2} a t^2$ ), you may need the quadratic formula! The complexity of the rearrangement depends on how the variable appears in the formula. To solve $P = \dfrac{V^2}{R}$ for V, first multiply both sides by R: $PR = V^2$ . Then take the square root: $V = \pm \sqrt{PR}$ . Choice B correctly isolates V through multiplication and taking the square root with ± to get $V = \pm \sqrt{PR}$ . A common distractor like choice A fails by not addressing the square and the division properly, resulting in an incorrect linear form; include the square root for the squared term. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. Example: $T = 2\pi \sqrt{\dfrac{L}{g}}$ solve for L: square both sides → $T^2 = 4\pi^2 \dfrac{L}{g}$ , multiply by g → $g T^2 = 4\pi^2 L$ , divide by $4\pi^2$ → $L = \dfrac{g T^2}{4\pi^2}$ . Each step undoes an operation! Watch for variables appearing multiple times: if your target variable appears in multiple places (like x in $xy + xz = w$ ), factor it out first: $x(y + z) = w$ , then $x = \dfrac{w}{y + z}$ . If you don't factor, you'll struggle to isolate! Also, when taking square roots of a variable, remember ± if the formula context allows both positive and negative (though often context restricts to positive only, like radius r ≥ 0). Physics and geometry formulas usually want positive values only!
In finance, compound interest can be modeled by $A = P(1+r)^t$ , where $A$ is final amount, $P$ is principal, $r$ is interest rate per period, and $t$ is the number of periods. Solve $A = P(1+r)^t$ for $P$ .
Answer: $P = \dfrac{A}{(1+r)^t}$ This question tests your ability to rearrange formulas to solve for a specific variable—essential for using formulas flexibly in science, engineering, and real-world problem-solving. Rearranging formulas with multiple variables works exactly like solving numeric equations, except the answer contains other variables instead of numbers: treat the variable you're solving for as the 'unknown x,' treat all other variables as 'known numbers,' then use inverse operations to isolate your target variable. Starting with A = $P(1 + r)^t$ , divide both sides by $(1 + r)^t$ to isolate P, resulting in P = A / $(1 + r)^t$ . Choice A correctly isolates P through division to get P = A / $(1 + r)^t$ . A distractor like choice B might confuse it with solving for A instead, but always check which variable is the target. The formula rearrangement recipe: (1) Identify what you're solving for (that's your 'x'), (2) Identify what operation(s) are being done to that variable in the original formula, (3) Apply inverse operations in reverse order to isolate it (just like numeric equations!), (4) Simplify the result—combine fractions, simplify radicals, etc. You're building strong finance skills—keep going!

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