Systems of Equations - PSAT Math

A small lab mixed two salt solutions. Solution A is $10\%$ salt and Solution B is $25\%$ salt. The lab wants $12$ liters of a mixture that is $20\%$ salt. Let $x$ be liters of A and $y$ be liters of B. What is the ordered pair $(x,y)$ that satisfies the system?
Answer: $\left(4,8\right)$ We need to find how many liters of each solution to mix, where Solution A is 10% salt, Solution B is 25% salt, and we want 12 liters of 20% salt mixture. The system is: x + y = 12 (total volume) and 0.10x + 0.25y = 0.20(12) (salt content). From the second equation: 0.10x + 0.25y = 2.4. I'll solve the first equation for x: x = 12 - y, then substitute: 0.10(12 - y) + 0.25y = 2.4. Expanding: 1.2 - 0.10y + 0.25y = 2.4, so 0.15y = 1.2 and y = 8. Therefore, x = 12 - 8 = 4, giving us (4, 8). A common error in mixture problems is forgetting to multiply the final concentration by the total volume. Always set up the salt equation as: (concentration A)(volume A) + (concentration B)(volume B) = (final concentration)(total volume).
The graph shown in the figure displays two lines in the xy-plane representing the system $\begin{cases} ax + 2y = 10 \\ 3x - by = 6 \end{cases}$ where $a$ and $b$ are constants. If the lines intersect at the point $(4, -1)$ , what is the value of $a + b$ ?
Answer: $-3$ Since the lines intersect at $(4, -1)$ , this point satisfies both equations. Substituting $(4, -1)$ into the first equation: $4a + 2(-1) = 10$ , so $4a - 2 = 10$ , which gives $4a = 12$ and $a = 3$ . Substituting $(4, -1)$ into the second equation: $3(4) - b(-1) = 6$ , so $12 + b = 6$ , which gives $b = -6$ . Therefore, $a + b = 3 + (-6) = -3$ .
Consider the system $\begin{cases}2x-3y=6\\4x-6y=12\end{cases}.$ How many ordered pairs (x, y) satisfy both equations?
Answer: infinitely many solutions When you encounter a system of linear equations, you're looking at how two lines relate to each other on a coordinate plane. They can intersect at one point, be parallel (never intersect), or be the same line entirely. Let's examine this system by looking at the relationship between the equations. Notice that if you multiply the first equation $2x - 3y = 6$ by 2, you get $4x - 6y = 12$ , which is exactly the second equation. This means both equations represent the same line—they're equivalent. When two equations in a system represent the same line, every point that satisfies one equation automatically satisfies the other. Since a line contains infinitely many points, there are infinitely many solutions to this system. You can verify this algebraically: from the first equation, $x = \frac{6 + 3y}{2}$ . Substituting any value for $y$ gives you a corresponding $x$ value that satisfies both equations. Answer choice (A) is wrong because exactly one solution occurs when two distinct lines intersect at a single point. Answer choice (B) is incorrect because no solution happens when you have parallel lines that never meet—this would occur if the equations had the same left side but different right sides (like $2x - 3y = 6$ and $2x - 3y = 8$ ). Answer choice (D) is wrong because systems of linear equations can only have zero, one, or infinitely many solutions—never a finite number greater than one. Study tip: When you see a system where one equation is a multiple of another, immediately check if they're equivalent—this signals infinitely many solutions.
The figure below shows the graphs of three lines: $\ell_1$ , $\ell_2$ , and $\ell_3$ . Lines $\ell_1$ and $\ell_2$ intersect at $(1, 4)$ ; lines $\ell_2$ and $\ell_3$ intersect at $(5, 2)$ ; and lines $\ell_1$ and $\ell_3$ intersect at $(-1, -2)$ . If the equation of $\ell_2$ is $y = mx + b$ , what is the value of $m + b$ ?
Answer: $4$ Line $\ell_2$ passes through $(1, 4)$ and $(5, 2)$ . The slope is $m = \frac{2-4}{5-1} = \frac{-2}{4} = -\frac{1}{2}$ . Using point-slope form with $(1, 4)$ : $y - 4 = -\frac{1}{2}(x - 1)$ , so $y = -\frac{1}{2}x + \frac{1}{2} + 4 = -\frac{1}{2}x + \frac{9}{2}$ . Therefore $m = -\frac{1}{2}$ and $b = \frac{9}{2}$ , giving $m + b = -\frac{1}{2} + \frac{9}{2} = \frac{8}{2} = 4$ .
A table shows values that satisfy a linear relationship between $x$ and $y$ : when $x=1$ , $y=5$ ; when $x=3$ , $y=1$ . Another equation is $x+y=6$ . If the first relationship and the equation $x+y=6$ are treated as a system, what is the solution $(x,y)$ ?
Answer: $\left(1,5\right)$ The question asks for the solution (x, y) to the system formed by the linear relationship from points (1,5) and (3,1), and the equation x + y = 6. To solve this system, I will use the substitution method. The line from the points has slope -2, so y = -2x + 7. Substitute into x + y = 6: x + (-2x + 7) = 6, simplify: -x + 7 = 6, subtract 7: -x = -1, x = 1, then y = 6 - 1 = 5. Thus, (1,5) satisfies both. A common error is miscalculating the slope or intercept from the points. When given points, find the equation first and verify with one point; for systems, check the solution in both equations.
A train travels 300 miles. It travels part of the trip at 60 mph and the rest at 75 mph. The total travel time is 4.4 hours. Let $x$ be miles traveled at 60 mph and $y$ be miles traveled at 75 mph. What is $(x,y)$ ?
Answer: $(120,180)$ The question asks for miles at each speed for 300 miles in 4.4 hours, 60 mph and 75 mph. Set up x + y =300, (x/60) + (y/75) =4.4. Use substitution or elimination; first solve for times. Multiply second equation by 300 to clear denominators: 5x +4y =1320. Wait, 300/60=5, 300/75=4, yes 5x +4y=4.4*300=1320. Now system x+y=300, 5x+4y=1320. Multiply first by 4: 4x+4y=1200. Subtract from second: (5x+4y)-(4x+4y)=1320-1200, x=120. Then y=180. (120,180). Choices A (120,180) yes. Marked A. Check: time 120/60=2, 180/75=2.4, total 4.4 yes. Good.
A jar contains only nickels ( $n$ ) and dimes ( $d$ ). The total number of coins is 38, and the total value is $$2.80. Which ordered pair $(n,d)$ represents the number of nickels and dimes in the jar?
Answer: $(20,18)$ The question asks for the ordered pair (n, d) representing the number of nickels and dimes. I will use the substitution method after setting up the system: n + d = 38 and 0.05n + 0.10d = 2.80 (or 5n + 10d = 280 in cents). Simplify the second to n + 2d = 56. Substitute n = 38 - d into it: 38 - d + 2d = 56, so 38 + d = 56, d = 18, then n = 20. Thus, (n, d) = (20, 18). A key error is forgetting to convert to cents or mishandling decimals. As a strategy, check the total value with the found numbers to verify.
A store sells notebooks for $3 each and pens for $2 each. A customer buys 17 items total and spends $43. Let $n$ be notebooks and $p$ be pens. Which ordered pair $(n,p)$ satisfies both conditions?
Answer: $(9,8)$ The question asks for the ordered pair (n, p) where n is notebooks at $3 each and p is pens at $2 each, with 17 items and $43 spent. Use the elimination method on n + p = 17 and 3n + 2p = 43 by multiplying the first by 2: 2n + 2p = 34. Subtract from the second: (3n + 2p) - (2n + 2p) = 43 - 34, so n = 9, then p = 8. A common error is switching n and p, leading to (8, 9). Clearly label variables and check the total cost and count.
A small lab has two solutions of salt water. Solution A is $10\%$ salt and Solution B is $25\%$ salt. The lab wants to make 40 liters of a mixture that is $16\%$ salt. Let $x$ be liters of A and $y$ be liters of B. What is $(x,y)$ ?
Answer: $(24,16)$ The question asks for (x, y) where x is liters of 10% salt solution A and y of 25% salt solution B to make 40 liters of 16% salt. Use the substitution method from x + y = 40, so x = 40 - y, and plug into 0.1x + 0.25y = 6.4: 0.1(40 - y) + 0.25y = 6.4. This simplifies to 4 - 0.1y + 0.25y = 6.4, so 0.15y = 2.4 and y = 16, then x = 24. A key error is miscalculating the salt amount as 0.16*40 = 6 instead of 6.4, leading to wrong values. In mixture problems, always verify the total volume and concentration in the solution.
A printing company charges $45 to set up a job plus $0.15 per flyer. Another company charges $30 to set up plus $0.20 per flyer. For how many flyers $f$ will the total costs be equal?
Answer: $300$ The question asks for the number of flyers where two printing companies' costs are equal, with setups $45 and $30, per flyer $0.15 and $0.20. Set up 45 + 0.15f = 30 + 0.20f, solve using elimination. Subtract 30 + 0.15f from both: 15 = 0.05f, f=15/0.05=300. Verify: first 45+0.15*300=45+45=90, second 30+0.20*300=30+60=90 yes. A key error is mixing up the per-unit costs, leading to wrong difference. Double-check calculations with decimals by multiplying through by 100 to use integers.
A student buys 5 items consisting of sandwiches and salads. Sandwiches cost $7 each and salads cost $5 each. The total cost is $31. Let $s$ be sandwiches and $a$ be salads. What is $(s,a)$ ?
Answer: $(3,2)$ The question asks for the number of sandwiches and salads, total 5 items for $31, sandwiches $7, salads $5. Set up s + a =5, 7s +5a =31. Substitution: s=5-a, 7(5-a)+5a=31, 35-7a+5a=31, 35-2a=31, -2a=-4, a=2. Then s=3. (3,2). Choices (1,4)(2,3)(3,2)(4,1). Marked C (3,2) yes. Check:3+2=5,7*3+5*2=21+10=31 yes. Good.
Solve the system using substitution: $y=2x-7$ and $3x+y=8$ . Be careful to substitute for $y$ in the second equation and then back-substitute to find $y$ . What is the solution $(x,y)$ to the system?
Answer: $\left(3,-1\right)$ We need to solve using substitution as requested, with y already isolated in the first equation. Substituting $y = 2x - 7$ into $3x + y = 8$ : $3x + (2x - 7) = 8$ , which simplifies to $5x - 7 = 8$ , so $5x = 15$ , giving $x = 3$ . Now back-substitute to find y: $y = 2(3) - 7 = 6 - 7 = -1$ . Therefore, the solution is $(3, -1)$ . The common error is forgetting to back-substitute or making arithmetic mistakes when combining like terms. Always verify by checking both original equations: $-1 = 2(3) - 7$ ✓ and $3(3) + (-1) = 8$ ✓.
The figure shows two lines $p$ and $q$ with equations $ax + by = 12$ and $2ax - by = 6$ , respectively, where $a$ and $b$ are positive constants. The lines intersect at $(3, 2)$ . What is the value of $ab$ ?
Answer: $6$ Adding the two equations: $3ax = 18 \Rightarrow ax = 6$ . With $x=3$ , $a=2$ . Substitute into first: $2(3) + b(2) = 12 \Rightarrow 2b = 6 \Rightarrow b = 3$ . Thus $ab = 6$ .
A gym charges a $30 sign-up fee plus $15 per month. Another gym charges a $10 sign-up fee plus $20 per month. After how many months $m$ will the total costs be equal?
Answer: $4$ The question asks for the number of months after which the total costs of two gyms are equal, given their sign-up fees and monthly charges. Set up the equations as $30 + 15m = 10 + 20m$ and solve using the elimination method. Subtract $10 + 15m$ from both sides to get $20 = 5m$ , then divide by 5 to find $m = 4$ . Verify by substituting $m = 4$ back: first gym costs $30 + 60 = 90$ , second costs $10 + 80 = 90$ , which matches. A key error is forgetting to subtract the sign-up fees correctly, leading to an incorrect m like 3 or 5. When setting up cost equations, always double-check the constants and coefficients before solving.
Two lines are shown on the coordinate plane. One line passes through $(-4,4)$ and $(4,-4)$ . The other line passes through $(-4,-2)$ and $(4,2)$ . What is the $x$ -coordinate of the solution to the system (the intersection point)?
Answer: $0$ The question requires the x-coordinate of the intersection of two lines given by points. First line: slope (-4-4)/(4+4) = -1, equation y = -x; second: slope (2+2)/(4+4) = 1/2, equation y = (1/2)x. Set equal: -x = (1/2)x, -x - (1/2)x = 0, - (3/2)x = 0, x = 0. Then y = 0, but only x is needed. Errors often occur in slope calculation, like ignoring negatives. Another pitfall is incorrect equation simplification. For efficiency, solve for intersection algebraically and confirm with one point.
Two lines are given by the equations $2x-3y=6$ and $4x-6y=12$ . A student claims the system has exactly one solution because the coefficients are different. How many solutions does the system actually have?
Answer: infinitely many This problem presents equations 2x - 3y = 6 and 4x - 6y = 12 and asks how many solutions exist. Notice that if we multiply the first equation by 2, we get 4x - 6y = 12, which is exactly the second equation. This means both equations represent the same line, so every point on this line is a solution - there are infinitely many solutions. The student's error was assuming different coefficients mean different lines, but proportional equations (where one is a multiple of the other) represent the same line. When two equations in a system are equivalent, the system has infinitely many solutions.
Consider the system of equations $\begin{cases}3x-2y=7\\6x-4y=14\end{cases}$ A student claims the system has one solution because there are two equations and two variables. How many solutions does this system actually have?
Answer: infinitely many The question asks how many solutions the system $3x - 2y = 7$ and $6x - 4y = 14$ has, despite a student's claim of one solution. To solve this system, I will use the elimination method. Multiply the first equation by 2: $6x - 4y = 14$ , which is identical to the second equation. This means the equations represent the same line, so they have infinitely many solutions. The student's error likely comes from assuming that two equations with two variables always yield exactly one solution, ignoring dependent systems. A key mistake is not checking if one equation is a multiple of the other. A useful test-taking strategy is to attempt elimination and see if you get $0 = 0$ , indicating infinite solutions.
A class collected donations in $5 and $10 bills only. They collected 38 bills total worth $275. Let $f$ be the number of $5 bills and $t$ be the number of $10 bills. What is $(f,t)$ ?
Answer: $(21,17)$ The question asks for the number of $5 bills (f) and $10 bills (t) given 38 total bills worth $275. We set up the system f + t = 38 and 5f + 10t = 275, using substitution by solving the first for f = 38 - t. Substituting into the second gives 5(38 - t) + 10t = 275, simplifying to 190 + 5t = 275, so 5t = 85 and t = 17; then f = 21. The pair (21, 17) satisfies both equations, as 21 + 17 = 38 and 5*21 + 10*17 = 275. A key error is mixing up the variables or forgetting to divide by 5 in solving for t. As a test-taking strategy, check the total value after finding the numbers to ensure it matches $275.
Solve the system: $5x+2y=4$ and $10x+4y=9$ . The second equation looks like a multiple of the first, but the constants differ. How many solutions does the system have?
Answer: $0$ We need to determine if this system has no solution, one solution, or infinitely many solutions. Notice that if we multiply the first equation by 2, we get 10x + 4y = 8, but the second equation is 10x + 4y = 9. This means we have 10x + 4y = 8 and 10x + 4y = 9 simultaneously, which is impossible since 8 ≠ 9. The equations represent parallel lines (same slope, different y-intercepts) that never intersect. The key insight is recognizing when the left sides of equations are proportional but the right sides aren't - this always indicates parallel lines with no solution.
Consider the system $3x-6y=12$ and $x-2y=4$ . The equations may represent the same line, parallel lines, or intersecting lines. How many solutions does this system have?
Answer: infinitely many We need to determine if these equations represent the same line, parallel lines, or intersecting lines. Let me rewrite the first equation in the same form as the second by dividing by 3: x - 2y = 4. This is exactly the same as the second equation x - 2y = 4. Since both equations are identical, they represent the same line, meaning every point on the line is a solution. The common mistake is not simplifying equations to check if they're equivalent. When two equations in a system are identical, the system has infinitely many solutions.

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