Match a system’s solutions to its graph. | Systems of equations in two variables | SAT Math

Graph questions still come down to the same intersection points you would find algebraically.

Core Idea

Each algebraic solution $(𝑥, 𝑦)$ is an intersection point on the graph. Solve the system to find the points, or read intersection points from the graph to identify the solutions.

Understanding

Solution = intersection point.

Solve the system algebraically to find all solution pairs.
Read the intersection points from the graph if the graph is given.
Match the exact thing the question asks for: point, quadrant, or one coordinate.

Pay attention to whether the prompt wants the point with the greater y-value or positive x.

Step by Step

Solve the system algebraically to find all solution pairs $(𝑥, 𝑦)$ .
Identify where these points fall on the coordinate plane.
If given a graph, read the intersection coordinates and verify in both equations.
Answer the specific graphical question (which point, which quadrant, which coordinate).

Misconceptions

Mixing up x-coordinates and y-coordinates when reporting an intersection point.
Assuming intersection points must have integer coordinates — they can be fractions or radicals.
Reading the wrong intersection point when the question specifies "greater y-coordinate" or "positive x."

Question

Worked Example

In the $𝑥 𝑦$ -plane, the line $𝑦 = 𝑥 - 1$ intersects the parabola $𝑦 = 𝑥 2 - 4 𝑥 + 3$ at two points. What is the $𝑥$ -coordinate of the intersection point with the greater $𝑦$ -coordinate?

Select an answer to see the explanation