Question

How do you graph
y = x + 2
y = 3x - 2

Answer 1

To graph the equations \(y = x + 2\) and \(y = 3x - 2\), follow these steps:

### Step 1: Graph \(y = x + 2\)

1. **Identify the Slope and Y-Intercept**:
- The equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- Here, the slope \(m = 1\) and the y-intercept \(b = 2\).

2. **Plot the Y-Intercept**:
- Start by plotting the point \((0, 2)\) on the graph.

3. **Use the Slope to Find Another Point**:
- From \((0, 2)\), use the slope to find another point. Since the slope is 1, move 1 unit up (rise) and 1 unit right (run) to get to \((1, 3)\).
- Plot this point.

4. **Draw the Line**:
- Draw a straight line through the points \((0, 2)\) and \((1, 3)\). Extend the line in both directions.

### Step 2: Graph \(y = 3x - 2\)

1. **Identify the Slope and Y-Intercept**:
- For this equation, the slope \(m = 3\) and the y-intercept \(b = -2\).

2. **Plot the Y-Intercept**:
- Plot the point \((0, -2)\) on the graph.

3. **Use the Slope to Find Another Point**:
- From \((0, -2)\), use the slope to find another point. Since the slope is 3, move 3 units up (rise) and 1 unit right (run) to get to \((1, 1)\).
- Plot this point.

4. **Draw the Line**:
- Draw a straight line through the points \((0, -2)\) and \((1, 1)\). Extend the line in both directions.

### Step 3: Find the Intersection Point

To find the intersection point of the two lines, set the equations equal to each other:

\[
x + 2 = 3x - 2
\]

1. Rearranging gives:

\[
2 + 2 = 3x - x
\]
\[
4 = 2x
\]
\[
x = 2
\]

2. Substitute \(x = 2\) back into either equation to find \(y\):

\[
y = 2 + 2 = 4
\]

So, the intersection point is \((2, 4)\).

### Summary of the Graphing Process:
- **First Line**: Start at \((0, 2)\), slope is 1, passes through \((1, 3)\).
- **Second Line**: Start at \((0, -2)\), slope is 3, passes through \((1, 1)\).
- **Intersection Point**: The lines intersect at \((2, 4)\).

Finally, make sure to label the lines and the intersection point on your graph for clarity!