Lesson 6-1: Solve Systems of Equations by Graphing

Property

A solution to a system of equations is an ordered pair $(x, y)$ that satisfies each equation in the system simultaneously.

To check whether an ordered pair is a solution, substitute the coordinates into each equation to verify that they result in true statements.

Examples

• Verifying a Solution: Is $(3, 7)$ a solution to the system $y = 2x + 1$ and $y = 4x - 5$?

Check equation 1: $7 = 2(3) + 1$ becomes 7 = 7 (True).
Check equation 2: $7 = 4(3) - 5$ becomes 7 = 7 (True).
Yes, it is the solution.

• The "One Line" Trap: Given the system $y = 2x + 1$ and $y = -x + 4$, the point $(2, 5)$ lies on the first line but not the second line because $5 \neq -2 + 4$. Therefore, it is NOT a solution to the system.

Property

To solve a system of linear equations by graphing, follow these steps:

1. Convert equations to slope-intercept form ($y = mx + b$) by isolating $y$.
2. Graph the first equation by plotting the y-intercept at $(0, b)$ and using the slope ($m$) to find the next point.
3. Graph the second equation on the same coordinate system.
4. Identify the point of intersection, which is the solution, and check it in both original equations.

Examples

• Converting First: Convert $3x + 2y = 8$ to slope-intercept form. Isolate $y$: $2y = -3x + 8$, so $y = -\frac{3}{2}x + 4$.
• Graphing to Solve: To solve the system $y = x + 1$ and $y = -x + 3$, we graph both lines. They intersect at the point $(1, 2)$. Checking this point in both equations confirms it is the solution.
• Special Lines: Remember that a horizontal line ($y = c$) has a slope of 0, and a vertical line ($x = c$) has an undefined slope. They are always perpendicular to each other.

Explanation

Many linear equations are not initially written in slope-intercept form, making it difficult to identify the slope and y-intercept directly. Once you use algebra to isolate $y$, graphing is like a simple treasure hunt. The solution to a system is the exact point where the two treasure paths cross.