Question

On a coordinate plane, a line with negative slope goes through (0, 4) and (4, 0).
The graph is shown for the equation y = –x + 4.

Which equation, when graphed with the given equation, will form a system that has an infinite number of solutions?
y = negative 4 (x + 1)
y = negative (x + 4)
y = negative one-half (2 x minus 8)
y = x + negative 4

Answer 1

To determine which equation will form a system with the equation \( y = -x + 4 \) that has an infinite number of solutions, we need to find an equation that represents the same line. This means the two equations should be equivalent.

The given equation is:

\[
y = -x + 4
\]

Now we will analyze the options provided:

1. **Option 1:** \( y = -4(x + 1) \)

This can be simplified:
\[
y = -4x - 4
\]
This is not equivalent to \( y = -x + 4 \).

2. **Option 2:** \( y = -(x + 4) \)

This can be simplified:
\[
y = -x - 4
\]
This is also not equivalent to \( y = -x + 4 \).

3. **Option 3:** \( y = -\frac{1}{2}(2x - 8) \)

This can be simplified:
\[
y = -\frac{1}{2}(2x) + \frac{1}{2}(8) = -x + 4
\]
This equation is equivalent to \( y = -x + 4 \).

4. **Option 4:** \( y = x - 4 \)

This can be simplified:
\[
y = x - 4
\]
This is also not equivalent to \( y = -x + 4 \).

The only equation that is equivalent to \( y = -x + 4 \) is Option 3:

\[
\text{y = } -\frac{1}{2}(2x - 8) \implies y = -x + 4.
\]

Thus, when this equation is graphed along with \( y = -x + 4 \), the system will have an infinite number of solutions.

The correct answer is:

**y = -\frac{1}{2}(2x - 8)**.