Question

fiona wrote the linear equation $y = \frac{2}{5}x - 5$. when henry wrote his equation, they discovered that his equation had all the same solutions as fiona’s. which equation could be henry’s?\
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$\bigcirc$ $x - \frac{5}{4}y = \frac{25}{4}$\
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$\bigcirc$ $x - \frac{5}{2}y = \frac{25}{4}$\
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$\bigcirc$ $x - \frac{5}{4}y = \frac{25}{2}$\
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$\bigcirc$ $x - \frac{5}{2}y = \frac{25}{2}$

Explanation:

Step1: Start with Fiona's equation

Fiona's equation is \( y = \frac{2}{5}x - 5 \). Let's rearrange this to the standard form \( Ax + By = C \).
First, multiply both sides by 5 to eliminate the fraction: \( 5y = 2x - 25 \).
Then, rearrange terms: \( 2x - 5y = 25 \). We can also divide both sides by 2 to simplify: \( x - \frac{5}{2}y = \frac{25}{2} \). Wait, no, let's do it step by step. Wait, from \( y=\frac{2}{5}x - 5 \), let's solve for \( x \) in terms of \( y \) or rearrange to standard form. Let's move \( \frac{2}{5}x \) to the left and \( y \) to the right: \( \frac{2}{5}x - y = 5 \). Now, multiply both sides by 5 to get \( 2x - 5y = 25 \). Now, divide both sides by 2: \( x - \frac{5}{2}y = \frac{25}{2} \). Wait, but let's check the options. Wait, maybe another approach. Let's take each option and solve for \( y \) and see if it matches Fiona's equation.

Step2: Analyze Option D ( \( x - \frac{5}{2}y = \frac{25}{2} \) )

Let's solve \( x - \frac{5}{2}y = \frac{25}{2} \) for \( y \).
Subtract \( x \) from both sides: \( -\frac{5}{2}y = -x + \frac{25}{2} \).
Multiply both sides by \( -\frac{2}{5} \): \( y = \frac{2}{5}x - 5 \), which is exactly Fiona's equation. Let's verify the other options to be sure.

Step3: Analyze Option A ( \( x - \frac{5}{4}y = \frac{25}{4} \) )

Solve for \( y \):
\( -\frac{5}{4}y = -x + \frac{25}{4} \)
Multiply by \( -\frac{4}{5} \): \( y = \frac{4}{5}x - 5 \). Not the same as Fiona's (which has slope \( \frac{2}{5} \)), so A is wrong.

Step4: Analyze Option B ( \( x - \frac{5}{2}y = \frac{25}{4} \) )

Solve for \( y \):
\( -\frac{5}{2}y = -x + \frac{25}{4} \)
Multiply by \( -\frac{2}{5} \): \( y = \frac{2}{5}x - \frac{5}{2} \). The y-intercept is \( -\frac{5}{2} \), but Fiona's is -5, so B is wrong.

Step5: Analyze Option C ( \( x - \frac{5}{4}y = \frac{25}{2} \) )

Solve for \( y \):
\( -\frac{5}{4}y = -x + \frac{25}{2} \)
Multiply by \( -\frac{4}{5} \): \( y = \frac{4}{5}x - 10 \), which is not Fiona's equation.

So the correct option is D.

Answer:

D. \( x - \frac{5}{2}y = \frac{25}{2} \)