1Cademy - Solving $$\{x + y = 2,\; x

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Solving a System of Linear Equations by Graphing

Example

Solving $\{x + y = 2,\; x - y = 4\}$ by Graphing

Solve the system $\left\{\begin{array}{l} x + y = 2 \\ x - y = 4 \end{array}\right.$ using the graphing method.

Because both equations are in standard form, use the intercept method to graph each line.

First equation: $x + y = 2$

x-intercept: Set $y = 0$ : $x + 0 = 2$ , so $x = 2$ . The x-intercept is $(2, 0)$ .
y-intercept: Set $x = 0$ : $0 + y = 2$ , so $y = 2$ . The y-intercept is $(0, 2)$ .

Second equation: $x - y = 4$

x-intercept: Set $y = 0$ : $x - 0 = 4$ , so $x = 4$ . The x-intercept is $(4, 0)$ .
y-intercept: Set $x = 0$ : $0 - y = 4$ , so $-y = 4$ and $y = -4$ . The y-intercept is $(0, -4)$ .

Plot both pairs of intercepts on the same coordinate system and draw the two lines. The lines intersect at the point $(3, -1)$ .

Verify the solution by substituting $x = 3$ and $y = -1$ into both original equations:

First equation: $3 + (-1) = 2$ . Since $2 = 2$ is true ✓
Second equation: $3 - (-1) = 3 + 1 = 4$ . Since $4 = 4$ is true ✓

Because $(3, -1)$ satisfies both equations, the solution of the system is $(3, -1)$ .

This example illustrates that when both equations in a system are in standard form with small integer coefficients, finding the x- and y-intercepts of each equation is a quick way to graph both lines without converting to slope-intercept form.

Updated 2026-04-21

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OpenStax Elementary Algebra

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