Lesson 2: Solve Systems by Graphing

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.

Property

A system of equations can be classified by the number of solutions, which is determined entirely by how the two lines relate to each other visually and algebraically:

• One Solution (Intersecting): The lines have different slopes. The system is consistent and independent.
• No Solution (Parallel): The lines have the exact same slope but different y-intercepts. The system is inconsistent.
• Infinite Solutions (Coincident): The lines have the same slope and the same y-intercept. The system is consistent and dependent.

Examples

• One Solution: The lines $y = x + 1$ and $y = -x + 3$ intersect at the point $(1, 2)$, giving exactly one solution.
• No Solution: The lines $y = 2x + 1$ and $y = 2x + 4$ are parallel (same slope, different y-intercepts) and never intersect.
• Infinite Solutions: The equations $y = 3x - 2$ and $6x - 2y = 4$ represent the same line when graphed, so every point on the line is a solution.

Explanation

When you graph two lines, they can only relate in three ways: they cross once, they never cross because they are parallel, or they are actually the exact same line. By simply looking at the 'm' and 'b' in their equations, you can instantly predict how many solutions the system has without even needing to draw the graph!