Systems of Linear Equations in Two Variables Pattern - Graphical Systems

This pattern gives you the graph of two lines and asks you to find the solution to the system. The solution is the point where the two lines intersect.

Reading the Intersection Point

The solution to a system of two linear equations is the ordered pair $(x, y)$ where both lines cross. Look at the graph, find where the lines meet, and read off the coordinates.

The graph of a system of linear equations is shown. What is the solution $(x, y)$ to the system?

A) $(-4, 3)$
B) $(0, 3)$
C) $(0, -1)$
D) $(-1, 0)$

The two lines intersect at the point where $x = -4$ and $y = 3$, so the solution is $(-4, 3)$. The answer is A. Option B is the $y$-intercept of the horizontal line. Options C and D are intercepts of the slanted line — not the intersection point.

The most common wrong answers are $y$-intercepts, $x$-intercepts, or points that lie on only one of the two lines. Always check that your chosen point sits on both lines.

The graph of a system of linear equations is shown in the $xy$-plane. What is the solution $(x, y)$ to the system?

A) $(1, 3)$
B) $(2, 4)$
C) $(4, 2)$
D) $(3, 5)$

The two lines cross at $(2, 4)$. The answer is B. Options A and D lie on one line only, and option C lies on the other line only.

The graph of a system of linear equations is shown. What is the solution $(x, y)$ to the system?

A) $(0, 1)$
B) $(5, 0)$
C) $(0, -5)$
D) $(2, -3)$

The lines intersect at $(2, -3)$. The answer is D. The other options are intercepts: $(0, 1)$ and $(0, -5)$ are $y$-intercepts and $(5, 0)$ is an $x$-intercept of the individual lines.

SPR Variant: Give Just the $x$- or $y$-Coordinate

Some problems show the same kind of graph but ask for only the $x$-coordinate or only the $y$-coordinate of the intersection point (student-produced response, no answer choices).

The graph of a system of linear equations is presented in the $xy$-plane. The solution is the point $(x, y)$. What is the value of $x$?

Find the intersection point on the graph. If the lines cross at $(-3, 4)$, the answer is $-3$.

The graph of a system of two linear equations is shown. If the solution is $(x, y)$, what is the value of $y$?

Read the intersection point from the graph. If the lines meet at $(6, 4)$, the answer is $4$.

Tip: Read both coordinates carefully — a common mistake is reporting the $y$-value when the question asks for $x$, or vice versa.

Identifying the System of Equations from a Graph

A harder variant shows two lines and asks which system of equations matches the graph. To solve this, determine the slope and $y$-intercept of each line, write the equations, and match to the choices.

In the $xy$-plane, the graph of a system of two linear equations is shown. Which of the following is the system of equations?

A) $5x + 5y = 20$ and $6x - 3y = 42$
B) $5x - 5y = 20$ and $6x + 3y = 42$
C) $-5x + 5y = 20$ and $3x + 6y = 42$
D) $5x + 5y = 50$ and $6x + 3y = 42$

Line 1 (upward slope): passes through $(4, 0)$ and $(6, 2)$. Slope $= \dfrac{2 - 0}{6 - 4} = 1$. Using point-slope form: $y - 0 = 1(x - 4)$, so $x - y = 4$. Multiply by 5: $5x - 5y = 20$.

Line 2 (downward slope): passes through $(6, 2)$ and $(7, 0)$. Slope $= \dfrac{0 - 2}{7 - 6} = -2$. Using point-slope form: $y - 2 = -2(x - 6)$, giving $2x + y = 14$. Multiply by 3: $6x + 3y = 42$.

The answer is B.

Adding a Third Equation to a Graphed System

The hardest variant shows two graphed lines and gives a third equation algebraically. You must determine how many solutions the combined three-equation system has.

Strategy: (1) Read the intersection point of the two graphed lines. (2) Substitute that point into the third equation. (3) If it satisfies the equation, the system has exactly one solution. If not, it has zero solutions.

The graph of a system of two linear equations is shown in the $xy$-plane. If the equation $3x + 2y = -4$ is added to the system, how many solutions $(x, y)$ would the new system of three equations have?

A) Zero
B) Exactly one
C) Exactly two
D) Infinitely many

The two graphed lines intersect at $(-2, 1)$. Check the third equation: $3(-2) + 2(1) = -6 + 2 = -4$. Since $-4 = -4$, the point satisfies the third equation. The three-equation system has exactly one solution. The answer is B.

The graph of a system of two linear equations is shown in the $xy$-plane. A third equation, $x - 2y = 5$, is combined with the graphed system. How many solutions does the resulting system of three equations have?

A) Zero
B) Exactly one
C) Exactly two
D) Infinitely many

The graphed lines intersect at $(2, -1)$. Substitute into the third equation: $(2) - 2(-1) = 2 + 2 = 4$. Since $4 \neq 5$, the point does not satisfy the third equation. The system has zero solutions. The answer is A.

Key insight: A system of three distinct lines can never have exactly two solutions. The answer is always zero, one, or infinitely many. If the two graphed lines intersect at a single point, just check whether that point is on the third line.