1Cademy - Solving $$\{y = 2x - 3,\; -6x + 3y = -9\}$$ by Graphing

Learn Before

Solving a System of Linear Equations by Graphing

Example

Solving $\{y = 2x - 3,\; -6x + 3y = -9\}$ by Graphing

Solve the system $\left\{\begin{array}{l} y = 2x - 3 \\ -6x + 3y = -9 \end{array}\right.$ using the graphing method.

First equation: $y = 2x - 3$ is already in slope-intercept form. The slope is $m = 2$ and the y-intercept is $(0, -3)$ . Graph this line using its slope and y-intercept.

Second equation: $-6x + 3y = -9$ is in standard form, so find its intercepts. Setting $x = 0$ gives $3y = -9$ , so $y = -3$ and the y-intercept is $(0, -3)$ . Setting $y = 0$ gives $-6x = -9$ , so $x = \frac{3}{2}$ and the x-intercept is $(\frac{3}{2}, 0)$ .

Graph both equations on the same coordinate system. The two lines land directly on top of each other — they are the same line.

Because every point on the line makes both equations true, there are infinitely many ordered pairs that satisfy both equations. The system has infinitely many solutions.

Converting the second equation to slope-intercept form confirms why the lines coincide: dividing each term of $-6x + 3y = -9$ by $3$ gives $-2x + y = -3$ , which simplifies to $y = 2x - 3$ — exactly the first equation. Both equations share slope $m = 2$ and y-intercept $(0, -3)$ , so they are coincident lines.

Updated 2026-04-21

Contributors are:

Who are from:

References

Learn Before

Related

Learn After