Middle School Math Quiz: Understand System Solutions As Intersections
20 questions · exam conditions
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Understand System Solutions As IntersectionsQuestion 1 of 20

Consider the system of equations:

{y=3x−4y=−x+8\begin{cases} y=3x-4 \\ y=-x+8 \end{cases}

Which ordered pair is the solution (the point where the two lines intersect)?

(3,5)(3,5)
(5,3)(5,3)
(2,2)(2,2)
(4,4)(4,4)
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Middle School Math Quiz

Middle School Math Quiz: Understand System Solutions As Intersections

Practice Understand System Solutions As Intersections in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Understand System Solutions As Intersections, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Consider the system of equations:

{y=3x−4y=−x+8\begin{cases} y=3x-4 \\ y=-x+8 \end{cases}

Which ordered pair is the solution (the point where the two lines intersect)?

  1. (3,5)(3,5) (correct answer)
  2. (5,3)(5,3)
  3. (2,2)(2,2)
  4. (4,4)(4,4)
Explanation: This question tests understanding that a system solution is the intersection point of graphs—the (x,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope different intercepts, no intersection/no solution), or same line (infinite points/infinite solutions). Point (x,y) is solution if substituting into both equations gives true statements (x=3, y=5 in y=3x-4: 5=3(3)-4=5✓, and y=-x+8: 5=-(3)+8=5✓, both true so (3,5) solves system). To verify, set 3x-4 = -x+8, yielding 4x=12 so x=3, then y=5, confirming (3,5) as the intersection. The correct choice is A, as C reverses coordinates, and B and D are points not on both lines. Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,y order matters), claiming parallel lines meet.

Question 2

Is the point (3,4)(3,4) a solution to this system (meaning it lies on both lines)?

{y=x+1y=2x−2\begin{cases} y=x+1\\ y=2x-2 \end{cases}
  1. No, because it makes the first equation true but not the second.
  2. No, because it makes neither equation true.
  3. No, because it makes the second equation true but not the first.
  4. Yes, because it makes both equations true. (correct answer)
Explanation: Tests understanding system solution is intersection point of graphs—the (x,y)(x,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope different intercepts, no intersection/no solution), or same line (infinite points/infinite solutions). Point (x,y)(x,y) is solution if substituting into both equations gives true statements (for x=2, y=5 in y=2x+1y=2x+1: 5=2(2)+1=55=2(2)+1=5 ✓, and y=−x+7y=-x+7: 5=−(2)+7=55=-(2)+7=5 ✓, both true so (2,5)(2,5) solves system). For this system, substituting (3,4) into y=x+1y=x+1 gives 4=3+14=3+1 (true), and into y=2x−2y=2x-2 gives 4=6−24=6-2 (true), so it lies on both lines. The point (3,4) is indeed the solution, making choice A correct. A common error is verifying only one equation and assuming it's the solution without checking the second. Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,y order matters), claiming parallel lines meet.

Question 3

How many solutions does this system have?

{y=3x+2y=3x−1\begin{cases} y = 3x + 2 \\ y = 3x - 1 \end{cases}

(Think about how many times the lines intersect.)

  1. Infinitely many solutions
  2. Exactly 2 solutions
  3. Exactly 1 solution
  4. No solutions (correct answer)
Explanation: Tests understanding system solution is intersection point of graphs—the (x,yx,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope 33 but different intercepts 22 and −1-1, no intersection/no solution), or same line (infinite points/infinite solutions). Point (x,yx,y) is solution if substituting into both equations gives true statements (for example, trying to set 3x+2=3x−13x+2=3x-1 yields 2=−12=-1, false, so no solution). The system has lines with same slope 33 but different intercepts 22 and −1-1, so they are parallel and do not intersect. The correct classification is no solutions, as parallel lines never cross. A common error is claiming they intersect despite same slope or wrong arithmetic in checking consistency. Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,yx,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,yx,y order matters), claiming parallel lines meet.

Question 4

Two linear equations form a system where one equation represents a line with slope m1=2m_1 = 2 and the other has slope m2=2m_2 = 2. Without graphing, what can be determined about the intersection of these lines?

  1. The lines will intersect at exactly one point since they have the same positive slope.
  2. The lines will either be parallel with no intersection or identical with infinite intersections. (correct answer)
  3. The lines will intersect at exactly two points because both slopes are equal to 2.
  4. The lines will be perpendicular and intersect at one point near the origin.
Explanation: When two lines have the same slope, they are either parallel (different y-intercepts, no solution) or the same line (same y-intercept, infinitely many solutions). They cannot intersect at exactly one point. Choice A incorrectly assumes same slope means one intersection. Choice C is impossible since two lines intersect at most once. Choice D confuses equal slopes with perpendicular lines (which have negative reciprocal slopes).

Question 5

Two lines are graphed on a coordinate plane. Line A passes through points (0,4)(0, 4) and (2,0)(2, 0), while Line B passes through points (0,−1)(0, -1) and (3,2)(3, 2). What does the intersection point of these lines represent?

  1. The solution to the system formed by the equations of both lines. (correct answer)
  2. The point where both lines have the same slope and y-intercept.
  3. The midpoint between the y-intercepts of the two linear equations.
  4. The point where the sum of both equations equals zero.
Explanation: The intersection point of two lines represents the solution to the system of linear equations formed by those lines. This point satisfies both equations simultaneously. Choice B is incorrect because lines with the same slope and y-intercept would be the same line. Choice C describes a calculation unrelated to intersection. Choice D misunderstands what intersection means algebraically.

Question 6

A student graphs the lines y=2x+1y=2x+1 and y=−x+7y=-x+7 on the same coordinate plane. What is the solution to the system (the intersection point of the two lines)?

  1. (2,5)(2,5) (correct answer)
  2. (3,7)(3,7)
  3. (2,1)(2,1)
  4. (5,2)(5,2)
Explanation: To solve the system, set the two expressions for yy equal and solve for xx: 2x+1=−x+72x+1=-x+7, so 3x=63x=6 and x=2x=2. Substituting back gives y=2(2)+1=5y=2(2)+1=5, so the intersection point is (2,5)(2,5). Checking with the other equation: y=−(2)+7=5y=-(2)+7=5, which matches. Choice B (3,73,7) is incorrect because substituting x=3x=3 into y=2x+1y=2x+1 gives y=7y=7, but y=−3+7=4y=-3+7=4, not 7. Choice C (2,12,1) has the right xx-value but a yy-value that satisfies neither equation. Choice D (5,25,2) reverses the coordinates of the correct answer.

Question 7

Two students are comparing a graph and an algebraic solution for the system:

{y=−2x+9y=x\begin{cases} y=-2x+9 \\ y=x \end{cases}

They claim the lines intersect at (3,3)(3,3). Which check correctly verifies that (3,3)(3,3) is the intersection (solution) of the system?

  1. Check that 3+3=93+3=9.
  2. Substitute into both: 3=−2(3)+93=-2(3)+9 and 3=33=3.
  3. Substitute into the second equation only: 3=33=3. (correct answer)
  4. Substitute into the first equation only: 3=−2(3)+93=-2(3)+9.
Explanation: This question tests understanding that a system solution is the intersection point of graphs—the (x,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope different intercepts, no intersection/no solution), or same line (infinite points/infinite solutions). Point (x,y) is solution if substituting into both equations gives true statements (x=3, y=3 in y=−2x+9y=-2x+9: 3=−2(3)+9=3✓3=-2(3)+9=3✓, and y=xy=x: 3=3✓3=3✓, both true so (3,3) solves system). Verification checks both equations for (3,3), confirming the intersection. The correct choice is C, as A and B check only one, D is unrelated arithmetic. Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,y order matters), claiming parallel lines meet.

Question 8

Two students are comparing a graph and an algebraic solution for the system:

{y=−2x+9y=x\begin{cases} y=-2x+9 \\ y=x \end{cases}

They claim the lines intersect at (3,3)(3,3). Which check correctly verifies that (3,3)(3,3) is the intersection (solution) of the system?

  1. Substitute into both: 3=−2(3)+93=-2(3)+9 and 3=33=3.
  2. Substitute into the second equation only: 3=33=3.
  3. Substitute into the first equation only: 3=−2(3)+93=-2(3)+9. (correct answer)
  4. Check that 3+3=93+3=9.
Explanation: This question tests understanding that a system solution is the intersection point of graphs—the (x,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope different intercepts, no intersection/no solution), or same line (infinite points/infinite solutions). Point (x,y) is solution if substituting into both equations gives true statements (x=3, y=3 in y=−2x+9y=-2x+9: 3=−2(3)+9=33=-2(3)+9=3✓, and y=xy=x: 3=33=3✓, both true so (3,3) solves system). Verification checks both equations for (3,3), confirming the intersection. The correct choice is C, as A and B check only one, D is unrelated arithmetic. Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,y order matters), claiming parallel lines meet.

Question 9

A student graphs the lines y=2x+1y=2x+1 and y=−x+7y=-x+7 on the same coordinate plane. What is the solution to the system (the intersection point of the two lines)?

  1. (3,7)(3,7)
  2. (2,5)(2,5) (correct answer)
  3. (5,2)(5,2)
  4. (2,1)(2,1)
Explanation: This question tests understanding that a system solution is the intersection point of graphs—the (x,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope different intercepts, no intersection/no solution), or same line (infinite points/infinite solutions). Point (x,y) is solution if substituting into both equations gives true statements (x=2, y=5 in y=2x+1: 5=2(2)+1=5✓, and y=-x+7: 5=-(2)+7=5✓, both true so (2,5) solves system). To verify, set 2x+1 = -x+7, yielding 3x=6 so x=2, then y=5, confirming (2,5) as the intersection. The correct choice is B, as A reverses coordinates, C has wrong y, and D is unrelated. Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,y order matters), claiming parallel lines meet.

Question 10

The graph displays a system where Line J intersects Line K at point (−2,1)(-2, 1). If a third line, Line L, is added to create the system {J,L}\{J, L\}, and this new system has no solution, what must be true about Line L's relationship to Line J?

  1. Line L must be perpendicular to Line J and pass through the origin point.
  2. Line L must have the same slope as Line J but a different y-intercept value. (correct answer)
  3. Line L must pass through point (−2,1)(-2, 1) and have twice the slope of Line J.
  4. Line L must be identical to Line J with the same slope and y-intercept values.
Explanation: For a system to have no solution, the lines must be parallel (same slope) but distinct (different y-intercepts). Choice A describes perpendicular lines, which would intersect at one point. Choice C describes a line through the intersection point, which would give one solution. Choice D describes identical lines, which would give infinitely many solutions.

Question 11

Based on the coordinate plane shown, what can be concluded about the system of equations represented by the two graphed lines?

  1. The system has no solution because the lines have different slopes.
  2. The system has infinitely many solutions because both lines pass through the origin.
  3. The system has exactly one solution at the point where the lines intersect. (correct answer)
  4. The system has no solution because the lines are parallel to each other.
Explanation: When two distinct lines intersect at exactly one point, the system has exactly one solution at that intersection point. Choice A incorrectly suggests different slopes mean no solution (actually the opposite). Choice B is wrong because passing through the origin doesn't create infinitely many solutions unless the lines are identical. Choice D would be correct only if the lines were parallel, which contradicts them intersecting.

Question 12

The graph shows three lines labeled P, Q, and R. If line P represents y=3x+1y = 3x + 1 and line Q represents y=−2x+6y = -2x + 6, what does the point where lines P and Q intersect tell us about the system formed by these equations?

  1. The intersection point is the unique solution that satisfies both equations simultaneously in the system. (correct answer)
  2. The intersection point shows that both equations have the same slope and therefore identical solutions.
  3. The intersection point indicates that the system has no solution because the lines are perpendicular.
  4. The intersection point proves that both equations are equivalent and represent the same mathematical relationship.
Explanation: The intersection point of two lines represents the unique ordered pair (x, y) that satisfies both equations in the system simultaneously. This is the definition of a solution to a system of linear equations. Choice B is incorrect because the lines have different slopes (3 and -2). Choice C wrongly suggests intersection means no solution. Choice D confuses intersection with identical equations.

Question 13

Looking at the coordinate plane, which statement correctly describes what the intersection point represents for the system of linear equations shown?

  1. The point represents the x-intercept where both lines cross the horizontal axis simultaneously.
  2. The point represents the ordered pair that makes both equations true when substituted for x and y. (correct answer)
  3. The point represents the location where both lines have equal slopes and parallel orientations.
  4. The point represents the midpoint between where each line crosses the coordinate axis boundaries.
Explanation: The intersection point of two lines represents the ordered pair (x, y) that, when substituted into both equations, makes both equations true simultaneously. This is the geometric representation of the algebraic solution. Choice A incorrectly focuses on x-intercepts only. Choice C describes parallel lines (which don't intersect). Choice D describes an irrelevant midpoint calculation.

Question 14

A system of equations is represented by two lines that intersect at (4,−3)(4, -3). Sarah incorrectly concludes that (4,−3)(4, -3) is a solution to only one of the equations. What is the error in Sarah's reasoning about intersection points?

  1. Sarah failed to recognize that intersection points satisfy both equations in the system simultaneously. (correct answer)
  2. Sarah confused the intersection point with the y-intercept of one of the individual equations.
  3. Sarah incorrectly calculated the intersection point and should have found a different coordinate pair.
  4. Sarah misunderstood what an intersection point represents geometrically, believing it belongs to only one line rather than both.
Explanation: By definition, an intersection point of two lines satisfies both equations simultaneously - that's what makes it an intersection point. Sarah's error was thinking it could satisfy only one equation. Choice B addresses a different type of error about intercepts. Choice C assumes the intersection point is wrong without evidence. Choice D incorrectly relates intersection points to infinite solutions.

Question 15

A student says the point (4,3)(4,3) is the solution to this system because it is on one of the lines:

{y=12x+1y=−x+7\begin{cases} y=\tfrac{1}{2}x+1 \\ y=-x+7 \end{cases}

Is (4,3)(4,3) a solution to the system (does it satisfy both equations)?

  1. Yes, because it makes y=12x+1y=\tfrac{1}{2}x+1 true.
  2. Yes, because it makes both equations true. (correct answer)
  3. No, because it makes both equations false.
  4. No, because it makes y=−x+7y=-x+7 false.
Explanation: This question tests understanding that a system solution is the intersection point of graphs—the (x,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope different intercepts, no intersection/no solution), or same line (infinite points/infinite solutions). Point (x,y) is solution if substituting into both equations gives true statements (x=4, y=3 in y=12x+1y=\frac{1}{2}x+1: 3=12(4)+1=33=\frac{1}{2}(4)+1=3 ✓, and y=−x+7y=-x+7: 3=−(4)+7=33=-(4)+7=3 ✓, both true so (4,3) solves system). Verification shows both equations hold true for (4,3), so it is the intersection point. The correct choice is A, as B checks only one, C incorrectly claims second false, and D says both false. Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,y order matters), claiming parallel lines meet.

Question 16

Maria graphs the system y=2x−1y = 2x - 1 and y=−x+5y = -x + 5 and finds they intersect at (2,3)(2, 3). She then graphs y=2x−1y = 2x - 1 and y=2x+4y = 2x + 4. How do the intersection properties of the second system compare to the first?

  1. The second system has infinitely many intersection points like the first system.
  2. The second system has exactly one intersection point like the first system.
  3. The second system has two intersection points, unlike the first system.
  4. The second system has no intersection points, unlike the first system. (correct answer)
Explanation: When you're comparing systems of linear equations, the key is to look at the slopes and y-intercepts to determine how many intersection points exist. Let's examine the second system: y=2x−1y = 2x - 1 and y=2x+4y = 2x + 4. Notice that both equations have the same slope (2) but different y-intercepts (-1 and +4). This means the lines are parallel - they run in exactly the same direction but are shifted vertically apart. Parallel lines never intersect, so this system has no solution points. This contrasts sharply with the first system, where y=2x−1y = 2x - 1 has slope 2 and y=−x+5y = -x + 5 has slope -1. Since the slopes are different, these lines intersect at exactly one point, which Maria correctly found to be (2, 3). Answer choice D correctly identifies that the second system has no intersection points, unlike the first system which has one. Answer A is wrong because the first system doesn't have infinitely many solutions - that only happens when you have the same line graphed twice. Answer B incorrectly assumes the second system has one intersection point like the first. Answer C suggests two intersection points, but two straight lines can intersect at most once (unless they're the same line). Remember this pattern: when two linear equations have the same slope but different y-intercepts, you're looking at parallel lines with no intersection. Different slopes mean exactly one intersection point.

Question 17

A student says the point (4,3)(4,3) is the solution to this system because it is on one of the lines:

{y=12x+1y=−x+7\begin{cases} y=\tfrac{1}{2}x+1 \\ y=-x+7 \end{cases}

Is (4,3)(4,3) a solution to the system (does it satisfy both equations)?

  1. No, because it makes both equations false.
  2. Yes, because it makes both equations true. (correct answer)
  3. Yes, because it makes y=12x+1y=\tfrac{1}{2}x+1 true.
  4. No, because it makes y=−x+7y=-x+7 false.
Explanation: This question tests understanding that a system solution is the intersection point of graphs—the (x,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope different intercepts, no intersection/no solution), or same line (infinite points/infinite solutions). Point (x,y) is solution if substituting into both equations gives true statements (x=4,y=3x=4, y=3 in y=12x+1y=\frac{1}{2}x+1: 3=12(4)+1=3✓3=\frac{1}{2}(4)+1=3\checkmark, and y=−x+7y=-x+7: 3=−4+7=3✓3=-4+7=3\checkmark, both true so (4,3)(4,3) solves system). Verification shows both equations hold true for (4,3)(4,3), so it is the intersection point. The correct choice is A, as B checks only one, C incorrectly claims second false, and D says both false. Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,y order matters), claiming parallel lines meet.

Question 18

Solve the system of equations. The solution is the intersection point of the two lines.

{y=2x+1y=−x+7\begin{cases} y=2x+1 \\ y=-x+7 \end{cases}
  1. (2,5)(2,5) (correct answer)
  2. (5,2)(5,2)
  3. (2,3)(2,3)
  4. (1,3)(1,3)
Explanation: Tests understanding system solution is intersection point of graphs—the (x,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope different intercepts, no intersection/no solution), or same line (infinite points/infinite solutions). Point (x,y) is solution if substituting into both equations gives true statements (x=2x=2, y=5y=5 in y=2x+1y=2x+1: 5=2(2)+1=5✓5=2(2)+1=5\checkmark, and y=−x+7y=-x+7: 5=−(2)+7=5✓5=-(2)+7=5\checkmark, both true so (2,5)(2,5) solves system). For this system, the lines y=2x+1y=2x+1 and y=−x+7y=-x+7 have different slopes (22 and −1-1), so they intersect at one point; solving 2x+1=−x+72x+1=-x+7 gives 3x=63x=6, x=2x=2, then y=5y=5. The correct intersection point is (2,5)(2,5), which is choice A. A common error is reversing the coordinates to (5,2)(5,2), but the point must satisfy both equations in the order (x,y). Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,y order matters), claiming parallel lines meet.

Question 19

A graph shows two lines that intersect at the point (3,−1)(3,-1). Which statement best explains why (3,−1)(3,-1) is the solution to the system?

  1. Because any point on either line is a solution to the system.
  2. Because (3,−1)(3,-1) is on both lines, so it makes both equations true at the same time. (correct answer)
  3. Because (3,−1)(3,-1) has a negative yy-value, so it must be the intersection.
  4. Because (3,−1)(3,-1) is the yy-intercept of both lines.
Explanation: This question tests understanding that a system solution is the intersection point of graphs—the (x,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope different intercepts, no intersection/no solution), or same line (infinite points/infinite solutions). The intersection at (3,-1) means it lies on both lines, satisfying both equations simultaneously. Verification assumes the graph shows distinct lines crossing at that point, classifying as one solution. The correct choice is B, as A misidentifies y-intercept, C irrelevant, D incorrect about any point. Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,y order matters), claiming parallel lines meet.

Question 20

Solve the system of equations. The solution represents the intersection point of the two lines.

{y=2x+1y=−x+7\begin{cases} y = 2x + 1 \\ y = -x + 7 \end{cases}

What is the solution (x,y)(x,y)?

  1. (2,3)(2,3)
  2. (5,2)(5,2)
  3. (2,5)(2,5) (correct answer)
  4. (3,2)(3,2)
Explanation: Tests understanding system solution is intersection point of graphs—the (x,y)(x,y) satisfying both equations simultaneously shown where lines cross. Two linear equations graph as two lines; solution is where lines intersect (point on both lines): one intersection (different slopes, one solution), parallel (same slope different intercepts, no intersection/no solution), or same line (infinite points/infinite solutions). Point (x,y)(x,y) is solution if substituting into both equations gives true statements (x=2, y=5 in y=2x+1y=2x+1: 5=2(2)+1=55=2(2)+1=5✓, and y=−x+7y=-x+7: 5=−(2)+7=55=-(2)+7=5✓, both true so (2,5)(2,5) solves system). The system has lines with slopes 2 and -1, different, so they intersect at one point, found by solving 2x+1=−x+72x+1=-x+7 yielding x=2x=2, y=5y=5. The correct solution is (2,5)(2,5), which is the intersection point. A common error is reversing coordinates to get (5,2) or misreading the point. Strategy: (1) graph both equations (or interpret given graph), (2) identify intersection (where lines cross, read coordinates), (3) verify algebraically (substitute x,y into both equations checking both true), (4) classify (intersecting once=one solution, parallel=no solution, same line=infinite). Connection: graph is visual (see solution location), algebra is exact (calculate precise coordinates), both show same information (intersection point = system solution). Mistakes: checking only one equation (must verify both), reading coordinates wrong (x,y order matters), claiming parallel lines meet.