Algebra 1 Practice - Solving Systems of Equations By Graphing (Example 2)

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

### 1. **Rewrite Each Equation in Slope-Intercept Form**
Ensure each equation is in the form:
\[
y = mx + b
\]
where \(m\) is the slope and \(b\) is the y-intercept.

### 2. **Graph Each Equation**
- Start with the y-intercept (\(b\)) and plot it on the y-axis.
- From that point, use the slope (\(m\)) to find other points. The slope is the ratio \(\frac{\text{rise}}{\text{run}}\), so move up/down and left/right accordingly.
- Draw a straight line through these points for each equation.

### 3. **Find the Intersection Point**
- The solution to the system is the point where the two lines intersect (if they do). This point represents the values of \(x\) and \(y\) that satisfy both equations.

### 4. **Interpret the Result**
- **One solution**: If the lines intersect at exactly one point, the system has one solution (consistent and independent).
- **No solution**: If the lines are parallel (same slope but different y-intercepts), the system has no solution (inconsistent).
- **Infinite solutions**: If the lines are the same (identical slopes and y-intercepts), the system has infinitely many solutions (consistent and dependent).

### Example:
Solve the system:
\[
\begin{aligned}
y &= 2x + 1 \\
y &= -x + 4
\end{aligned}
\]
1. The first equation is already in slope-intercept form: \( y = 2x + 1 \), with slope \(m = 2\) and y-intercept \(b = 1\).
2. The second equation is also in slope-intercept form: \( y = -x + 4 \), with slope \(m = -1\) and y-intercept \(b = 4\).
3. Graph both lines:
- For \( y = 2x + 1 \), start at (0, 1) and use the slope to plot points.
- For \( y = -x + 4 \), start at (0, 4) and use the slope to plot points.
4. The lines intersect at (1, 3), so the solution is \(x = 1, y = 3\).

This method visually shows the relationship between the equations, helping you find their common solution.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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