Question

On a coordinate plane, a line goes through (negative 4, 0) and (0, 2).
The graph of the equation y = 1
2
x + 2 is displayed. Which equations would intersect the orange line at the y-intercept? Check all that apply.
y = 7x + 2
y = 3x + 5
y = 2 minus StartFraction 7 Over 3 EndFraction x
y – 2 = 4x
3y = –9x – 6

Answer 1

To determine which equations intersect with the orange line (the line represented by \(y = \frac{1}{2}x + 2\)) at the y-intercept, we first identify the y-intercept of the equation of the orange line. The y-intercept occurs when \(x = 0\).

For the equation \(y = \frac{1}{2}x + 2\):

- Set \(x = 0\):
\[ y = \frac{1}{2}(0) + 2 = 2 \]

So, the y-intercept of the orange line is at the point (0, 2).

Next, we will find the y-intercepts of the other equations you provided by substituting \(x = 0\) into each one and checking if the resulting value of \(y\) equals 2.

1. **Equation: \(y = 7x + 2\)**
- When \(x = 0\):
\[ y = 7(0) + 2 = 2 \]
- This intersects at (0, 2).

2. **Equation: \(y = 3x + 5\)**
- When \(x = 0\):
\[ y = 3(0) + 5 = 5 \]
- This does not intersect.

3. **Equation: \(y = 2 - \frac{7}{3}x\)**
- When \(x = 0\):
\[ y = 2 - \frac{7}{3}(0) = 2 \]
- This intersects at (0, 2).

4. **Equation: \(y - 2 = 4x\)**
- Rearranging gives \(y = 4x + 2\).
- When \(x = 0\):
\[ y = 4(0) + 2 = 2 \]
- This intersects at (0, 2).

5. **Equation: \(3y = -9x - 6\)**
- Rearranging gives \(y = -3x - 2\).
- When \(x = 0\):
\[ y = -3(0) - 2 = -2 \]
- This does not intersect.

### Summary:
The equations that intersect with the orange line at the y-intercept (0, 2) are:
- \(y = 7x + 2\)
- \(y = 2 - \frac{7}{3}x\)
- \(y - 2 = 4x\)

These are the equations that apply.