1Cademy - Solving $$\{y = 6,\; 2x + 3y = 12\}$$ by Graphing

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Solving a System of Linear Equations by Graphing

Example

Solving $\{y = 6,\; 2x + 3y = 12\}$ by Graphing

Solve the system $\left\{\begin{array}{l} y = 6 \\ 2x + 3y = 12 \end{array}\right.$ using the graphing method.

The first equation, $y = 6$ , contains only the variable $y$ , so its graph is a horizontal line whose y-intercept is $6$ .

The second equation, $2x + 3y = 12$ , is in standard form, so it is most conveniently graphed using the intercept method. Setting $x = 0$ gives $3y = 12$ , so $y = 4$ and the y-intercept is $(0, 4)$ . Setting $y = 0$ gives $2x = 12$ , so $x = 6$ and the x-intercept is $(6, 0)$ .

Graph both lines on the same coordinate system. The horizontal line $y = 6$ and the line through $(0, 4)$ and $(6, 0)$ intersect at the point $(-3, 6)$ .

Verify the solution by substituting $x = -3$ and $y = 6$ into both original equations:

First equation: $6 \stackrel{?}{=} 6$ . True ✓
Second equation: $2(-3) + 3(6) = -6 + 18 = 12$ . Since $12 = 12$ is true ✓

Because $(-3, 6)$ satisfies both equations, the solution of the system is $(-3, 6)$ .

This example illustrates solving a system in which one equation is a horizontal line. Recognizing that $y = 6$ represents a horizontal line eliminates the need for any algebraic rewriting or point-plotting for that equation — its graph can be drawn immediately.

Updated 2026-04-21

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OpenStax Elementary Algebra

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