Solving Systems by Graphing
Solve systems of linear equations by graphing both equations and identifying the intersection point. Understand the limitations of this method (non-integer solutions) and how to check graphical answers algebraically.
A lesson of Algebra 1, in Systems of Equations & Inequalities, about 30 minutes.
Solving by Graphing
By the end of this lesson, you will be able to solve a system by drawing both lines and reading the crossing point, check the answer, and say when graphing is the right method and when it is not.
Draw Both, Read the Crossing
The method is exactly what the last lesson described.
- Put both equations into $y = mx + b$
- Graph each: dot at $b$, step by $m$
- Read the coordinates where they cross
- Check the pair in both original equations
Step 4 is not optional here, and it is not the same as checking your arithmetic. Graphing produces a READING, and a reading can be a little off. Substituting back either confirms the answer exactly or tells you the crossing was not on a lattice point.
Example 1: A Clean Crossing
Here is the whole method on one system, worked all the way through. $$\begin{cases} y = x + 1 \ y = -2x + 7 \end{cases}$$ Both are already in the right form, so start graphing.
- What is the y-intercept of the first line? 1
- What is the y-intercept of the second? 7
- The lines cross where their y values agree. Set x + 1 equal to -2x + 7 and solve for x 2
- Put x = 2 into either equation. What is y? 3
- Write the solution as a coordinate pair (2, 3)
Plot It: The Two Lines
Worked Example 1: Rearranging First
$$\begin{cases} 2x + y = 8 \ y = x - 1 \end{cases}$$
What the Picture Tells You Instantly
Before solving anything, the slopes say which of the three outcomes you are in.
| Slopes | Intercepts | Picture | Solutions |
|---|---|---|---|
| Different | anything | Cross once | One |
| Same | Different | Parallel | None |
| Same | Same | One line | Infinitely many |
$$y = 2x + 1 \text{ and } y = 2x - 4 \quad \text{parallel, no solution}$$
$$y = 2x + 1 \text{ and } 2y = 4x + 2 \quad \text{the same line, infinitely many}$$
That second pair is worth care. The two equations look different and are not: halving the second gives the first exactly. Compare after rearranging, never before.
Sort It: How Many Solutions
Look at each pair and decide, without solving.
When Graphing Is the Wrong Tool
Graphing is the method that SHOWS you what is happening, and it is the least precise one.
<!-- SLATE_IF_STRONG: math.algebra.systems-substitution --> <!-- END_SLATE_IF_STRONG -->