Non Linear Equations in One Variable and System of Equations in Two Variables Pattern - Interpret Graphs

Interpreting Graphs to Solve Systems

This pattern shows you a graph with two functions plotted in the $xy$-plane and asks you to find where they intersect. The solution to the system is the intersection point — the $(x, y)$ coordinates where both graphs pass through the same spot. No algebra needed; just read the graph carefully.

The Core Idea

The solution to a system of equations is the point where the graphs of both equations meet. If the graphs intersect at $(3, 5)$, then $x = 3$ and $y = 5$ satisfies both equations simultaneously.

Worked Examples

Example 1. The graph of a system of one linear equation and one quadratic equation is shown. Which ordered pair is the solution?

A) $(5, 2)$
B) $(4, 1)$
C) $(2, 5)$
D) $(4, 0)$

Look for where the line and the parabola cross. They intersect at $(5, 2)$.
Gotcha: Option B $(4, 1)$ is the vertex of the parabola — an important feature of the quadratic, but not the intersection with the line. Option C $(2, 5)$ swaps the $x$ and $y$ coordinates. Option D $(4, 0)$ is the $x$-intercept of the parabola.
The answer is A.

Example 2. The graphs of two equations are shown. What is the solution $(x, y)$?

A) $(5, 4)$
B) $(1, 2)$
C) $(4, 5)$
D) $(1, 0)$

The radical function and the horizontal line intersect at $(5, 4)$.
Gotcha: Option B $(1, 2)$ is the starting point of the radical function. Option C $(4, 5)$ swaps the coordinates.
The answer is A.

Example 3. A circle and a horizontal line intersect at two points. What is the solution for which $x > 0$?

A) $(5, 0)$
B) $(-4, 3)$
C) $(0, 5)$
D) $(4, 3)$

The circle and line intersect at $(-4, 3)$ and $(4, 3)$. The question asks for $x > 0$, so pick $(4, 3)$.
Gotcha: When there are two intersection points, read the constraint carefully. "$x > 0$" means the right-side intersection. "$x < 0$" would mean the left-side one.
The answer is D.

What to Do on Test Day

• Intersection = solution. The answer is always the point where both graphs cross each other, not a vertex, intercept, or other feature of a single graph.
• Don't swap coordinates. If the intersection is at $(5, 2)$, the answer is $(5, 2)$, not $(2, 5)$. The first number is always $x$ (horizontal), the second is $y$ (vertical).
• Watch for multiple intersections. If two graphs cross at two points, the question will add a constraint like "$x > 0$" or "the solution with the greater $y$-value" to narrow it down.
• Common distractors: Vertices of parabolas, $x$-intercepts, $y$-intercepts, and starting points of radical functions are all wrong answers. They are features of one graph, not of the system.
• Read carefully from the grid. Use the gridlines to identify exact coordinates. If the intersection appears between grid lines, look for the answer that matches a fractional value.