Question

tomas wrote the equation $y = 3x + \frac{3}{4}$. when sandra wrote her equation, they discovered that her equation had all the same solutions as tomas’s equation. which equation could be sandra’s?\
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\\(\bigcirc\\) $-6x + y = \frac{3}{2}$\
\
\\(\bigcirc\\) $6x + y = \frac{3}{2}$\
\
\\(\bigcirc\\) $-6x + 2y = \frac{3}{2}$\
\
\\(\bigcirc\\) $6x + 2y = \frac{3}{2}$

Explanation:

Step1: Analyze the original equation

Tomas's equation is \( y = 3x+\frac{3}{4} \). We can rewrite it in standard form \( Ax + By = C \) as \( - 3x+y=\frac{3}{4} \).

Step2: Check each option

• Option 1: \( -6x + y=\frac{3}{2} \). If we divide the original equation \( - 3x+y=\frac{3}{4} \) by \( \frac{1}{2} \), we get \( -6x + 2y=\frac{3}{2} \), not \( -6x + y=\frac{3}{2} \), so this is incorrect.
• Option 2: \( 6x + y=\frac{3}{2} \). The slope of Tomas's equation is \( 3 \), and the slope of this equation is \( - 6 \), which is different, so they are not the same line, incorrect.
• Option 3: \( -6x + 2y=\frac{3}{2} \). Divide the original equation \( - 3x+y=\frac{3}{4} \) by \( \frac{1}{2} \):

\[

$$\begin{align*} \frac{-3x}{\frac{1}{2}}+\frac{y}{\frac{1}{2}}&=\frac{\frac{3}{4}}{\frac{1}{2}}\\ -6x + 2y&=\frac{3}{2} \end{align*}$$

\]
This is the same as the given equation in option 3, so this is correct.

• Option 4: \( 6x + 2y=\frac{3}{2} \). The slope of this equation is \( - 3 \), which is different from the slope of Tomas's equation (\( 3 \)), so they are not the same line, incorrect.

Answer:

\( -6x + 2y=\frac{3}{2} \) (the third option)