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.

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). Both lines have slope 1 but different y-intercepts (2 and -3), so they are parallel and do not intersect. Verification confirms no common point satisfies both, classifying as no solutions. The correct choice is C, as A assumes intersection, B and D misclassify the type. 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.

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: $$y = 2x - 1$$ and $$y = 2x + 4$$. 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 $$y = 2x - 1$$ has slope 2 and $$y = -x + 5$$ 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.

When you encounter problems about systems of linear equations, think about what the different types of solutions tell you about how the lines relate to each other graphically.

If a system has exactly one solution at $$(3, -2)$$, this means there's precisely one point where both equations are satisfied simultaneously. Graphically, this represents the intersection point of two lines. Since linear equations graph as straight lines, and we know these lines meet at exactly one point, they must be two distinct lines that cross each other once.

Choice D correctly describes this situation - two different lines intersecting at exactly one point. This is the most common scenario for linear systems and represents what mathematicians call a "consistent and independent" system.

Choice A describes parallel lines, which never intersect. This would mean the system has no solution at all, contradicting the given information that there is a solution at $$(3, -2)$$.

Choice B describes the same line graphed twice, which would create infinitely many solutions (every point on the line satisfies both equations). This contradicts having exactly one solution.

Choice C mentions perpendicular lines intersecting at the origin, but this is too specific. While the lines could be perpendicular, they don't have to be, and we know they intersect at $$(3, -2)$$, not the origin $$(0, 0)$$.

Remember this pattern: one solution means two distinct intersecting lines, no solution means parallel lines, and infinitely many solutions means the same line repeated.

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).

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.

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.

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 $y=\frac{1}{2}x+1$: $3=\frac{1}{2}(4)+1=3$ ✓, and $y=-x+7$: $3=-(4)+7=3$ ✓, 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.

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=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). The second equation 2y=4x+2 simplifies to y=2x+1, which is the same as the first, so they represent the same line with infinite intersection points. The system has infinitely many solutions, making choice C correct. A common error is not simplifying the second equation and thinking the slopes differ. 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.

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 ($x=2$, $y=5$ in $y=2x+1$: $5=2(2)+1=5\checkmark$, and $y=-x+7$: $5=-(2)+7=5\checkmark$, both true so $(2,5)$ solves system). For this system, the lines $y=2x+1$ and $y=-x+7$ have different slopes ($2$ and $-1$), so they intersect at one point; solving $2x+1=-x+7$ gives $3x=6$, $x=2$, then $y=5$. The correct intersection point is $(2,5)$, which is choice A. A common error is reversing the coordinates to $(5,2)$, 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.