Question

A coordinate grid with a line labeled y equals StartFraction 2 over 5 EndFraction x minus 5 passing through the points (negative 5, negative 7) and (0, negative 5)
Fiona wrote the linear equation y = y equals StartFraction 2 over 5 EndFraction x minus 5.x – 5. When Henry wrote his equation, they discovered that his equation had all the same solutions as Fiona’s. Which equation could be Henry’s?

x – x minus StartFraction 5 over 4 EndFraction y equals StartFraction 25 over 4 EndFraction.y =
x – x minus StartFraction 5 over 2 EndFraction y equals StartFraction 25 over 4 EndFraction.y =
x – x minus StartFraction 5 over 4 EndFraction y equals StartFraction 25 over 2 EndFraction.y =
x – x minus StartFraction 5 over 2 EndFraction y equals StartFraction 25 over 2 EndFraction.y =

Answer 1

To find an equivalent equation to Fiona’s, we need to start with her equation and manipulate it into a different form without changing its solutions. Fiona’s equation is:

\[
y = \frac{2}{5}x - 5
\]

To find a different form of this equation, we can rearrange it into standard form, which is often given as:

\[
Ax + By = C
\]

Let's rearrange Fiona's equation:

1. Move \(y\) to one side:
\[
y - \frac{2}{5}x = -5
\]

2. Multiply each term by 5 to eliminate the fraction:
\[
5y - 2x = -25
\]

3. Rearranging gives:
\[
2x - 5y = 25
\]

Now we can identify the form that could belong to Henry. To find potential equivalent equations, we’ll look to manipulate \(2x - 5y = 25\) into the given choices.

1. Start with \(2x - 5y = 25\).
2. Divide through by a non-zero constant does not change the solution set.

So let’s check the four options:

- **Option 1:** \(x - \frac{5}{4}y = \frac{25}{4}\)
- **Option 2:** \(x - \frac{5}{2}y = \frac{25}{4}\)
- **Option 3:** \(x - \frac{5}{4}y = \frac{25}{2}\)
- **Option 4:** \(x - \frac{5}{2}y = \frac{25}{2}\)

This corresponds to each option in the format \(Ax + By = C\):

1. For **Option 1:** Multiply by 4:
\[
4x - 5y = 25 \text{, which is equivalent after manipulation.}
\]

2. For **Option 2:** Multiply by 4:
\[
4x - 10y = 25 \text{, not equivalent.}
\]

3. For **Option 3:** Multiply by 2:
\[
2x - 5y = 25 \text{, which is exactly the same.}
\]

4. For **Option 4:** Multiply by 2 gives:
\[
2x - 5y = 50 \text{, not equivalent.}
\]

So the correct answer, which is equivalent to Fiona's original equation, is **Option 3**:

\[
x - \frac{5}{4}y = \frac{25}{2}
\]