Lesson 5: Solving Equations by Graphing

Property

For equations with variables on both sides of the equation, begin as we did above—choose a “variable” side and a “constant” side, and then use the subtraction and addition properties of equality to collect all variables on one side and all constants on the other side.

Examples

• Solve $10x = 9x - 7$. Subtract $9x$ from both sides to get all the variables on the left, which gives the solution $x = -7$.

• Solve $3y - 8 = 6y$. Subtract $3y$ from both sides to gather the variables on the right, which gives $-8 = 3y$. Divide by 3 to get $y = -\frac{8}{3}$.

Property

Every point on the graph of an equation represents a solution to that equation. A solution to a system of two equations must be a point that lies on both graphs. Therefore, a solution to the system is a point where the two graphs intersect.

Examples

• For the system $y = x + 2$ and $y = -x + 4$, graphing the lines shows they intersect at the point $(1, 3)$. Therefore, the solution to the system is $x=1, y=3$.

• To solve the system $y = 1.7x + 0.4$ and $y = 4.1x + 5.2$, we can graph both lines. The intersection point occurs at $(-2, -3)$, which is the solution to the system.