Algebra · Learn by Concept

Algebra Help: Solving Systems Of Linear Equations

Review real example questions for Solving Systems Of Linear Equations in Algebra.

Question 1 / 10

0 of 10 answered

Is $(3,2)$ a solution to the system $$ \begin{cases} x+y=5\ 2x-y=4 \end{cases}

All questions

Question 1

Is $(3,2)$ a solution to the system $$ \begin{cases} x+y=5\ 2x-y=4 \end{cases}

Yes, because it satisfies both equations. (correct answer)
No, because it satisfies neither equation.
No, because it satisfies only $x+y=5$ .
No, because it satisfies only $2x-y=4$ .

Question 2

A coffee shop sells two types of drinks. Let $x$ represent the number of lattes sold and $y$ represent the number of cappuccinos sold. If the shop sold twice as many lattes as cappuccinos, and the total number of drinks sold was 36, which system of equations represents this situation?

$\begin{cases} x = 2y \\ x + y = 36 \end{cases}$ (correct answer)
$\begin{cases} y = 2x \\ x + y = 36 \end{cases}$
$\begin{cases} x + y = 2 \\ x = 36y \end{cases}$
$\begin{cases} 2x = y \\ x - y = 36 \end{cases}$

Explanation: The problem states that the shop sold twice as many lattes as cappuccinos, which means the number of lattes ( $x$ ) equals 2 times the number of cappuccinos ( $y$ ), so $x = 2y$ . The total number of drinks sold was 36, so $x + y = 36$ . Choice B incorrectly states that cappuccinos equal twice the lattes. Choice C has incorrect relationships entirely. Choice D incorrectly represents the total as a difference rather than a sum.

Question 3

Sarah is solving $\begin{cases} 3x + 2y = 16 \\ x - y = 2 \end{cases}$ and finds that $x = 4$ . She wants to check her work by substituting back into both original equations. If $x = 4$ is correct, what value of $y$ should make both equations true?

$y = 1$
$y = 4$
$y = 3$
$y = 2$ (correct answer)

Explanation: When you're checking solutions to systems of equations, you need to substitute your values into both original equations to verify they work. This is a crucial step that catches calculation errors and confirms your solution is correct. Since Sarah found $x = 4$ , let's find the corresponding $y$ value. Using the second equation $x - y = 2$ , substitute $x = 4$ : $4 - y = 2$ . Solving for $y$ : $y = 4 - 2 = 2$ . Now verify this works in both equations. First equation: $3(4) + 2(2) = 12 + 4 = 16$ ✓. Second equation: $4 - 2 = 2$ ✓. Both equations are satisfied, confirming $y = 2$ . Looking at the wrong answers: Choice A gives $y = 1$ . Checking the first equation: $3(4) + 2(1) = 14 \neq 16$ , so this fails. Choice B gives $y = 4$ . In the second equation: $4 - 4 = 0 \neq 2$ , which doesn't work. Choice C gives $y = 3$ . In the first equation: $3(4) + 2(3) = 18 \neq 16$ , and in the second: $4 - 3 = 1 \neq 2$ , so this fails both equations. The answer is D) $y = 2$ . Study tip: Always check your solutions in both original equations, not just one. A value might satisfy one equation by coincidence but fail the other, indicating an error in your work. This verification step is essential for catching mistakes in systems problems.

Question 4

Maria is solving the system $\begin{cases} 2x - 3y = 7 \\ 4x + y = 1 \end{cases}$ by elimination. She multiplies the first equation by 2 to get $4x - 6y = 14$ , then subtracts the second equation from this result. What equation does she obtain?

$-7y = 13$ (correct answer)
$-7y = 15$
$7y = -13$
$7y = -15$

Explanation: Maria multiplies the first equation by 2: $2(2x - 3y) = 2(7)$ gives $4x - 6y = 14$ . Then she subtracts the second equation $(4x + y = 1)$ from this result: $(4x - 6y) - (4x + y) = 14 - 1$ . This gives $4x - 6y - 4x - y = 13$ , which simplifies to $-7y = 13$ .

Question 5

When solving the system $\begin{cases} 5x - 2y = 8 \\ 3x + 4y = 22 \end{cases}$ using substitution, Alex solves the first equation for $y$ and gets $y = \frac{5x - 8}{2}$ . When he substitutes this into the second equation, what equation in terms of $x$ only does he obtain?

$3x + 4(\frac{5x - 8}{2}) = 22$
$3x + 2(5x - 8) = 22$
$13x - 16 = 22$ (correct answer)
$23x - 32 = 22$

Explanation: Alex correctly solved for $y$ : from $5x - 2y = 8$ , we get $y = \frac{5x - 8}{2}$ . Substituting into the second equation: $3x + 4(\frac{5x - 8}{2}) = 22$ . Simplifying: $3x + 4 \cdot \frac{5x - 8}{2} = 3x + 2(5x - 8) = 3x + 10x - 16 = 13x - 16 = 22$ . Choice A shows the substitution step but hasn't simplified. Choice B shows an incorrect simplification of the fraction. Choice D results from an arithmetic error in combining like terms.

Question 6

The graphs of the lines $y = -x + 6$ and $y = x + 2$ intersect at what point?

$(2, 4)$ (correct answer)
$(4, 2)$
$(1, 5)$
$(3, 3)$

Explanation: This question tests your ability to solve systems of linear equations—finding the (x, y) pair that makes both equations true at the same time. A system of equations is like a puzzle where you need to find values that work for both equations simultaneously: the solution (x, y) must make the first equation true AND make the second equation true. Graphically, this is where the two lines intersect—that one point where both equations are satisfied! To find the intersection, set -x + 6 = x + 2, so -x - x = 2 - 6, -2x = -4, x = 2; then y = 2 + 2 = 4. Choice A is correct because it gives the (2, 4) pair that satisfies both equations when you substitute back to check: 4 = -2 + 6 and 4 = 2 + 2. A distractor like Choice B (4, 2) might come from swapping x and y, but it fails because 2 ≠ -4 + 6? Wait, 2 = -4 + 6 is true, but 2 ≠ 4 + 2, so it only works for one. Here's how to choose a method: if one equation is already solved for a variable (like y = 3x + 1), use substitution—it's set up perfectly! If the coefficients of one variable are opposites (like 2x and -2x) or the same (like 3y and 3y), use elimination—one variable will cancel nicely. And you can always graph both lines to see where they cross! With practice, you'll spot the easiest method for each system. Always check your answer by plugging both x and y into BOTH original equations. If you get true statements (like 5 = 5 and 7 = 7), you're correct! If even one equation doesn't work, there's an error somewhere. This check habit catches almost all mistakes and builds confidence!

Question 7

How many solutions does the system have? $$ \begin{cases} 2x + 4y = 10 \ x + 2y = 5 \end{cases}

Exactly two solutions
No solution
Exactly one solution
Infinitely many solutions (correct answer)

Explanation: This question tests your ability to solve systems of linear equations—finding the (x, y) pair that makes both equations true at the same time. A system of equations is like a puzzle where you need to find values that work for both equations simultaneously: the solution (x, y) must make the first equation true AND make the second equation true. Graphically, this is where the two lines intersect—that one point where both equations are satisfied! Let's examine these equations closely. If we divide the first equation by 2, we get: $x + 2y = 5$ . But wait—that's exactly the same as the second equation! This means both equations represent the same line, so every point on that line is a solution. Choice C is correct because when two equations represent the same line, there are infinitely many solutions—every point on the line satisfies both equations. If you thought there was no solution, you might have confused this with parallel lines; these are the same line, not parallel lines! Here's how to choose a method: if one equation is already solved for a variable (like $y = 3x + 1$ ), use substitution—it's set up perfectly! If the coefficients of one variable are opposites (like $2x$ and $-2x$ ) or the same (like $3y$ and $3y$ ), use elimination—one variable will cancel nicely. And you can always graph both lines to see where they cross! With practice, you'll spot the easiest method for each system.

Question 8

The sum of two numbers is 15 and their difference is 3. Let $x$ be the larger number and $y$ be the smaller number.

\begin{cases} x + y = 15\\ x - y = 3 \end{cases}

What are the values of $x$ and $y$ ?

$(x, y) = (6, 9)$
$(x, y) = (9, 6)$ (correct answer)
$(x, y) = (12, 3)$
$(x, y) = (3, 12)$

Explanation: This question tests your ability to solve systems of linear equations—finding the (x, y) pair that makes both equations true at the same time. The elimination method (also called addition method) works by adding or subtracting the equations to make one variable disappear: if you have x + y = 5 and x - y = 1, adding them gives 2x = 6 because the y terms cancel out. Then solve for x, and use that to find y! We have x + y = 15 and x - y = 3. Adding these equations: (x + y) + (x - y) = 15 + 3, which gives 2x = 18, so x = 9. Substituting x = 9 into x + y = 15: 9 + y = 15, so y = 6. Since x is the larger number and y is the smaller number, we have x = 9 and y = 6. Choice B is correct because (x, y) = (9, 6) satisfies both equations: 9 + 6 = 15 ✓ and 9 - 6 = 3 ✓. Choice A would give 6 + 9 = 15 ✓ but 6 - 9 = -3 ≠ 3, which violates the condition that x is the larger number. Always check your answer by plugging both x and y into BOTH original equations. If you get true statements (like 5 = 5 and 7 = 7), you're correct! If even one equation doesn't work, there's an error somewhere. This check habit catches almost all mistakes and builds confidence!

Question 9

Based on the graph shown, what is the solution to the system of linear equations?

$(-1, 2)$
$(2, -1)$
$(1, -2)$ (correct answer)
$(-2, 1)$

Explanation: The solution to a system of linear equations graphically is the point where the two lines intersect. Looking at the graph, the two lines intersect at the point $(1, -2)$ . This means $x = 1$ and $y = -2$ satisfy both equations simultaneously.

Question 10

The sum of two numbers is $15$ and their difference is $3$ (larger minus smaller). What are the two numbers?

$9$ and $6$ (correct answer)
$12$ and $3$
$8$ and $7$
$10$ and $5$

Explanation: This question tests your ability to solve systems of linear equations—finding the (x, y) pair that makes both equations true at the same time. For word problems: first define your variables clearly ('let x = larger number, y = smaller number'), then translate each sentence into an equation. The word 'and' usually signals you have two separate equations that form your system! From the problem: x + y = 15 (sum) and x - y = 3 (difference). Using elimination by adding the equations: (x + y) + (x - y) = 15 + 3, which gives us 2x = 18, so x = 9. Substituting back: 9 + y = 15, so y = 6. The two numbers are 9 and 6. Choice A is correct because 9 + 6 = 15 ✓ and 9 - 6 = 3 ✓, satisfying both conditions. Choice B would give us 8 + 7 = 15 ✓ but 8 - 7 = 1, not 3, so it only satisfies one condition. Always check your answer by plugging both values back into the original context to ensure they make sense!