Question

(a)
Graph and then find the solution for
y=2x+1 and y= -3/2x + 8

Answer 1

To find the solution for the equations \(y = 2x + 1\) and \(y = -\frac{3}{2}x + 8\), we can graph both lines and then determine where they intersect.

### Step 1: Graph Each Equation

**Equation 1:** \(y = 2x + 1\)

- **Slope (m)**: 2
- **Y-intercept (b)**: 1
- **Points to plot**:
- When \(x = 0\), \(y = 2(0) + 1 = 1\) → Point (0, 1)
- When \(x = 1\), \(y = 2(1) + 1 = 3\) → Point (1, 3)
- When \(x = -1\), \(y = 2(-1) + 1 = -1\) → Point (-1, -1)

**Equation 2:** \(y = -\frac{3}{2}x + 8\)

- **Slope (m)**: \(-\frac{3}{2}\)
- **Y-intercept (b)**: 8
- **Points to plot**:
- When \(x = 0\), \(y = -\frac{3}{2}(0) + 8 = 8\) → Point (0, 8)
- When \(x = 2\), \(y = -\frac{3}{2}(2) + 8 = 5\) → Point (2, 5)
- When \(x = 4\), \(y = -\frac{3}{2}(4) + 8 = 2\) → Point (4, 2)

### Step 2: Draw the Graph

You can sketch the lines based on the points provided.

1. For \(y = 2x + 1\), draw a line that passes through the points (0, 1), (1, 3), and (-1, -1).
2. For \(y = -\frac{3}{2}x + 8\), draw a line that passes through the points (0, 8), (2, 5), and (4, 2).

### Step 3: Find the Intersection Point

To find the intersection point (solution), set the two equations equal to each other:

\[
2x + 1 = -\frac{3}{2}x + 8
\]

### Step 4: Solve for \(x\)

1. First, eliminate the fraction by multiplying everything by 2:

\[
4x + 2 = -3x + 16
\]

2. Move all \(x\) terms to one side and constant terms to the other side:

\[
4x + 3x = 16 - 2
\]

\[
7x = 14
\]

3. Divide by 7:

\[
x = 2
\]

### Step 5: Substitute \(x\) back to find \(y\)

Use either original equation to find \(y\). Let's use \(y = 2x + 1\):

\[
y = 2(2) + 1 = 4 + 1 = 5
\]

### Conclusion: Solution

The solution to the system of equations is the point of intersection:
\[
\boxed{(2, 5)}
\]

This means that the two lines intersect at the point (2, 5).