Solving Systems by Substitution - Math 1

A linear system has the property that when solved by substitution, one of the variables is eliminated completely, resulting in the equation $0 = 0$ . This means the system has:
Answer: infinitely many solutions because the equations represent the same line When substitution leads to $0 = 0$ , this is always true, indicating that the original equations are equivalent and represent the same line. Therefore, every point on that line is a solution, giving infinitely many solutions. Choice A describes a typical system with one solution, choice B describes an inconsistent system (which would give $0 = k$ where $k ≠ 0$ ), and choice D is impossible since two different lines can intersect at most once.
Consider the system where the first equation is $x = 3y + 1$ and the second equation becomes $2(3y + 1) + 5y = 23$ after substitution. What was the original second equation before substitution?
Answer: $2x + 5y = 23$ When you encounter a substitution problem, you're working backwards from the result to find the original equation. The key is understanding that substitution means replacing one variable with an equivalent expression from another equation. You're given that $x = 3y + 1$ and that after substitution, the second equation becomes $2(3y + 1) + 5y = 23$ . To find the original equation, you need to reverse the substitution process. Since $x = 3y + 1$ , wherever you see $(3y + 1)$ in the substituted equation, that's where $x$ originally appeared. Looking at $2(3y + 1) + 5y = 23$ , you can see that $(3y + 1)$ was substituted for $x$ . Replacing it back gives you $2x + 5y = 23$ , which is answer B. Answer A ( $6y + 2 + 5y = 23$ ) shows what you'd get if you distributed the 2 in the substituted equation, but this isn't the original form before substitution occurred. Answer C ( $2x + 5x = 23$ ) incorrectly suggests that both terms contained $x$ , but the $5y$ term remained unchanged during substitution. Answer D ( $x + 5y = 23$ ) is missing the coefficient 2 that appears in front of the substituted expression. Remember: when reversing substitution, look for the complete expression that was replaced (including any coefficients in front of it) and substitute the original variable back in its place.
Marcus is solving the system $\begin{cases} x + 2y = 10 \\ 3x + y = 11 \end{cases}$ by substitution. He isolates $x$ from the first equation and substitutes into the second equation, obtaining $3(10 - 2y) + y = 11$ . What is his next step to find $y$ ?
Answer: Simplify to get $30 - 6y + y = 11$ , then solve $-5y = -19$ From $x + 2y = 10$ , Marcus gets $x = 10 - 2y$ . Substituting: $3(10 - 2y) + y = 11$ becomes $30 - 6y + y = 11$ , which simplifies to $30 - 5y = 11$ , giving $-5y = -19$ . Choice B incorrectly combines $-6y + y$ as $-5y$ in the expansion step. Choice C uses 13 instead of 30 when distributing 3. Choice D has the wrong sign when moving terms.
The system $\begin{cases} 4x + y = 15 \\ 2x - 3y = -7 \end{cases}$ has solution $(x, y) = (a, b)$ . If instead we solve the equivalent system $\begin{cases} y = 15 - 4x \\ 2x - 3y = -7 \end{cases}$ by substitution, what equation in $x$ results?
Answer: $2x - 3(15 - 4x) = -7$ When solving systems of equations by substitution, you're replacing one variable with an expression from another equation. This method requires careful attention to which equation you substitute into. The key insight here is understanding what "substitution" means. You start with the equivalent system where the first equation is already solved for $y$ : $y = 15 - 4x$ . To use substitution, you take this expression for $y$ and substitute it into the second equation wherever you see $y$ . The second equation is $2x - 3y = -7$ . When you substitute $y = 15 - 4x$ into this equation, you replace the $y$ with $(15 - 4x)$ , giving you: $2x - 3(15 - 4x) = -7$ . This matches answer choice D. Let's see why the other options are wrong. Choice A ( $14x - 45 = -7$ ) shows what you'd get after expanding and simplifying the substitution, but the question asks for the equation that "results" from substitution, meaning the immediate result before simplification. Choice B ( $4x + (15 - 4x) = -7$ ) incorrectly substitutes into the first equation instead of the second. Choice C ( $2x - 45 + 12x = -7$ ) appears to be a partially expanded version with sign errors. Study tip: In substitution problems, always substitute the solved expression into the equation that still contains that variable. Double-check that you're using the correct equation and maintaining proper parentheses around the substituted expression.
The system $\begin{cases} 2x - y = 5 \\ 6x - 3y = 12 \end{cases}$ is being solved by substitution. From the first equation, $y = 2x - 5$ . After substituting into the second equation, what conclusion can be drawn?
Answer: The system has no solution since substitution leads to $0 = -3$ Substituting $y = 2x - 5$ into $6x - 3y = 12$ : $6x - 3(2x - 5) = 12$ , which gives $6x - 6x + 15 = 12$ , simplifying to $15 = 12$ or $0 = -3$ . This contradiction means no solution exists. Choice A is wrong because the equations aren't equivalent. Choice C assumes a solution exists. Choice D correctly identifies no solution but gives an incomplete geometric reason.
Consider the system $2x + 5y = 25$ and $x = y + 2$ . When this system is solved by substitution, what is the value of $y$ ?
Answer: $3$ Substituting $x = y + 2$ into $2x + 5y = 25$ : $2(y + 2) + 5y = 25$ . Expanding: $2y + 4 + 5y = 25$ , so $7y + 4 = 25$ , giving $7y = 21$ , thus $y = 3$ . Choice A would result from solving $7y = 7$ , choice B from $7y = 14$ , and choice D from $7y = 28$ .
When solving the system $5x - 2y = 4$ and $y = 3x + 1$ by substitution, a student gets the equation $5x - 2(3x + 1) = 4$ . After correctly expanding and simplifying, what equation should they solve next?
Answer: $-x - 2 = 4$ Expanding $5x - 2(3x + 1) = 4$ : $5x - 6x - 2 = 4$ , which simplifies to $-x - 2 = 4$ . Choice B has the wrong sign on the constant term, choice C incorrectly adds instead of subtracting the $x$ terms ( $5x + 6x$ ), and choice D forgets the negative sign when combining $5x - 6x$ .
Two equations are given: $x - 2y = 8$ and $3x + ky = 6$ , where $k$ is unknown. When solved by substitution using $x = 2y + 8$ , the second equation becomes $3(2y + 8) + ky = 6$ . If this simplifies to $(6 + k)y = -18$ , and the system has the solution $y = -2$ , what is the value of $k$ ?
Answer: $k = 3$ This problem tests your understanding of systems of linear equations and how substitution works to find unknown parameters. When you substitute $x = 2y + 8$ into the second equation $3x + ky = 6$ , you get $3(2y + 8) + ky = 6$ . Expanding this: $6y + 24 + ky = 6$ , which simplifies to $(6 + k)y = 6 - 24 = -18$ . So the simplified form is $(6 + k)y = -18$ . Since the system has the solution $y = -2$ , you can substitute this value: $(6 + k)(-2) = -18$ . Solving for $k$ : $-2(6 + k) = -18$ , so $-12 - 2k = -18$ . Adding 12 to both sides: $-2k = -6$ , therefore $k = 3$ . Let's examine why the other answers are incorrect. Choice A ( $k = -3$ ) would give you $(6 + (-3))y = 3y = -18$ , making $y = -6$ , not $y = -2$ . Choice B ( $k = 6$ ) would result in $(6 + 6)y = 12y = -18$ , so $y = -\frac{3}{2}$ , which doesn't match. Choice C ( $k = 9$ ) gives $(6 + 9)y = 15y = -18$ , making $y = -\frac{6}{5}$ , also incorrect. When working with systems involving unknown parameters, always use the given solution values to create equations you can solve for those parameters. The key is recognizing that if you know what $y$ equals, you can work backwards to find any unknowns in your simplified equation.
The system $3x - 2y = 0$ and $y = 2x - 1$ is solved by substitution. What is the value of $x + y$ ?
Answer: $5$ Substituting $y = 2x - 1$ into $3x - 2y = 0$ : $3x - 2(2x - 1) = 0$ . Expanding: $3x - 4x + 2 = 0$ , so $-x + 2 = 0$ , giving $x = 2$ . Then $y = 2(2) - 1 = 3$ . Therefore $x + y = 2 + 3 = 5$ . Choice A represents $xy$ , choice C represents $3x$ , and choice D represents $2x + 1$ .
A system has equations $y = 2x - 5$ and $3x + 4y = 22$ . When the substitution method is used, what equation results after substitution but before simplification?
Answer: $3x + 4(2x - 5) = 22$ When you encounter a system of equations problem asking about substitution "before simplification," you need to identify the exact moment after substitution occurs but before any algebraic simplification takes place. The substitution method involves taking one equation solved for a variable and substituting that expression into the other equation. Here, the first equation $y = 2x - 5$ is already solved for $y$ . To use substitution, you take this expression for $y$ and replace every $y$ in the second equation $3x + 4y = 22$ . Substituting $y = 2x - 5$ into $3x + 4y = 22$ gives you $3x + 4(2x - 5) = 22$ . This is the equation immediately after substitution but before any simplification—you haven't distributed the 4 or combined like terms yet. Choice A, $y + 4(2x - 5) = 22$ , incorrectly keeps the $y$ term instead of replacing it with $3x$ . Choice B, $3x + 2x - 5 = 22$ , shows partial simplification where someone incorrectly multiplied $4 \times 2x$ but forgot to multiply $4 \times (-5)$ . Choice D, $3x + 8x - 20 = 22$ , shows the equation after complete simplification (distributing the 4), which goes beyond what the question asks for. The correct answer is C because it shows the exact state after substitution but before any distributive property or combining of terms. Study tip: When a question asks for results "before simplification," stop immediately after the substitution step—don't distribute or combine terms.
A system of equations is given by $3x - 2y = 7$ and $y = mx + 4$ . After substitution, the resulting equation in $x$ is $3x - 2(mx + 4) = 7$ . If this simplifies to $-5x - 8 = 7$ , what is the value of $m$ ?
Answer: $m = 4$ Expanding $3x - 2(mx + 4) = 7$ gives $3x - 2mx - 8 = 7$ , which rearranges to $(3 - 2m)x - 8 = 7$ . For this to equal $-5x - 8 = 7$ , we need $3 - 2m = -5$ . Solving: $-2m = -8$ , so $m = 4$ . Choice A results from solving $3 - 2m = -5$ incorrectly as $m = -4$ . Choice B results from setting $3 + 2m = -5$ . Choice D results from solving $2m - 3 = 5$ .
The system $\begin{cases} 2x + y = 8 \\ 4x - 3y = 6 \end{cases}$ is being solved by substitution. From the first equation, $y = 8 - 2x$ . After substituting this into the second equation and solving, which ordered pair represents the solution?
Answer: $(3, 2)$ Substituting $y = 8 - 2x$ into $4x - 3y = 6$ : $4x - 3(8 - 2x) = 6$ , which gives $4x - 24 + 6x = 6$ , so $10x = 30$ and $x = 3$ . Then $y = 8 - 2(3) = 2$ . The solution is $(3, 2)$ . Choice B results from solving $10x = 20$ instead of $10x = 30$ . Choice C results from setting $y = 0$ and solving only the first equation. Choice D results from an arithmetic error giving $x = 1$ .
A student solving $\begin{cases} y = x + 3 \\ 2x - y = -1 \end{cases}$ by substitution writes: " $2x - (x + 3) = -1$ , so $2x - x + 3 = -1$ , giving $x = -4$ ." What error did the student make?
Answer: Failed to distribute the negative sign when substituting $y = x + 3$ The student wrote $2x - (x + 3) = -1$ but then incorrectly simplified it as $2x - x + 3 = -1$ . The correct simplification should be $2x - x - 3 = -1$ , giving $x - 3 = -1$ and $x = 2$ . The student failed to distribute the negative sign. Choice B is wrong because the isolation was correct. Choice C is wrong because that's not the arithmetic error made. Choice D is wrong because the substitution was into the correct equation.
A system of equations $ax + by = c$ and $x = dy + e$ is solved by substitution. After substituting the second equation into the first and collecting like terms, the resulting equation has the form $(ad + b)y + ae = c$ . For the system $6x + 7y = 50$ and $x = 2y + 3$ , what is the coefficient of $y$ in the resulting equation?
Answer: $19$ Using the formula $(ad + b)y + ae = c$ with $a = 6$ , $b = 7$ , $d = 2$ , and $e = 3$ : the coefficient of $y$ is $ad + b = 6(2) + 7 = 12 + 7 = 19$ . Choice A represents $ad + b - 2$ , choice B represents $ad + b - 1$ , and choice D represents $ad + b + 1$ . We can verify by direct substitution: $6(2y + 3) + 7y = 50$ becomes $12y + 18 + 7y = 50$ , or $19y + 18 = 50$ .
The system $10x + 3y = 29$ and $x + y = 7$ is solved by substitution. If you solve for $y$ in the second equation first, what simplified equation do you get after substitution?
Answer: $7x + 21 = 29$ From $x + y = 7$ , we get $y = 7 - x$ . Substituting into $10x + 3y = 29$ : $10x + 3(7 - x) = 29$ . Expanding: $10x + 21 - 3x = 29$ , which simplifies to $7x + 21 = 29$ . Choice A would come from $4x + 3y = 29$ , choice C would come from $16x + 3y = 29$ , and choice D shows the equation before combining like terms.
The system $3x + 4y = 26$ and $x - y = 1$ is to be solved by substitution. If you solve the second equation for $x$ first, what expression would you substitute into the first equation?
Answer: $y + 1$ When solving a system of equations by substitution, you need to isolate one variable in terms of the other, then substitute that expression into the remaining equation. Starting with the second equation $x - y = 1$ , you solve for $x$ by adding $y$ to both sides: $x = y + 1$ . This expression for $x$ is what you'll substitute into the first equation $3x + 4y = 26$ , replacing every $x$ with $(y + 1)$ . The correct answer is D) $y + 1$ because this is exactly the expression we derived when solving $x - y = 1$ for $x$ . Let's see why the other choices are wrong. Choice A) $x + 1$ doesn't make sense because you're trying to eliminate $x$ , not create an expression that still contains it. Choice B) $y - 1$ represents a sign error - if you incorrectly subtracted $y$ from both sides instead of adding it, you'd get $x = 1 - y$ , but that's not this expression. Choice C) $1 - y$ is what you'd get if you solved for $x$ incorrectly by moving the $y$ term to the right side without changing its sign. Remember that substitution requires you to express one variable completely in terms of the other. Always double-check your algebra when isolating variables, and make sure your final expression contains only the variable you're keeping, plus any constants.
A student attempts to solve the system $4x + y = 10$ and $x - 2y = 3$ by substitution. From the second equation, they solve for $x$ and get $x = 3 + 2y$ . When they substitute this into the first equation, what coefficient will $y$ have after simplification?
Answer: $9$ Substituting $x = 3 + 2y$ into $4x + y = 10$ : $4(3 + 2y) + y = 10$ . Expanding: $12 + 8y + y = 10$ , which simplifies to $12 + 9y = 10$ . The coefficient of $y$ is 9. Choice A would result from $4(2y) - y$ , choice B from forgetting to add the $y$ term, and choice D from incorrectly expanding $4(3 + 2y)$ as $12 + 11y$ .
Consider the system $2x - 3y = 6$ and $y = x - 4$ . When solved by substitution, what is the solution?
Answer: $(6, 2)$ When you encounter a system of linear equations like this, substitution is often the most efficient method when one equation is already solved for a variable. Here, the second equation gives you $y = x - 4$ , making it perfect for substitution. To solve by substitution, replace $y$ in the first equation with the expression from the second equation. Substituting $y = x - 4$ into $2x - 3y = 6$ gives you: $2x - 3(x - 4) = 6$ . Distribute the $-3$ : $2x - 3x + 12 = 6$ . Combine like terms: $-x + 12 = 6$ . Solving for $x$ : $-x = -6$ , so $x = 6$ . Now substitute back into $y = x - 4$ to get $y = 6 - 4 = 2$ . The solution is $(6, 2)$ . Let's check why the other answers are incorrect. Choice A) $(12, 8)$ would result from errors in distributing or combining like terms—perhaps forgetting the negative sign when distributing $-3$ . Choice B) $(8, 4)$ might come from solving $-x = -6$ incorrectly or making arithmetic mistakes in the substitution step. Choice C) $(9, 5)$ could result from sign errors or incorrectly setting up the substitution equation. Always verify your solution by substituting both values back into both original equations. For $(6, 2)$ : $2(6) - 3(2) = 12 - 6 = 6$ ✓ and $2 = 6 - 4$ ✓. When solving systems by substitution, work methodically through each step and double-check your arithmetic—small errors compound quickly in multi-step problems.
The system $x + 3y = 8$ and $2x - y = 2$ is solved by substitution. What is the solution?
Answer: $(2, 2)$ From $x + 3y = 8$ , we get $x = 8 - 3y$ . Substituting into $2x - y = 2$ : $2(8 - 3y) - y = 2$ . Expanding: $16 - 6y - y = 2$ , so $16 - 7y = 2$ , giving $7y = 14$ , thus $y = 2$ . Then $x = 8 - 3(2) = 2$ . Choice B gives $x + 3y = 10$ , choice C gives $2x - y = 5$ , and choice D gives $x + 3y = 4$ .
A system of equations is given by $2x + 3y = 11$ and $x = 4y - 5$ . When solving this system by substitution, Maria substituted the expression for $x$ into the first equation and obtained $2(4y - 5) + 3y = 11$ . After simplifying, what equation did she get before solving for $y$ ?
Answer: $11y - 10 = 11$ After substituting $x = 4y - 5$ into $2x + 3y = 11$ , Maria gets $2(4y - 5) + 3y = 11$ . Distributing: $8y - 10 + 3y = 11$ . Combining like terms: $11y - 10 = 11$ . Choice B shows the equation before combining like terms, choice C has the wrong sign on 10, and choice D incorrectly combines $8y + 3y = 5y$ .

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