Question
A coordinate grid with one line labeled y equals 3 x plus StartFraction 3 over 4 EndFraction. The line passes through points at (negative 1, negative 2) and (0, 1).
Tomas wrote the equation y = 3x +y equals 3 x plus StartFraction 3 over 4 EndFraction.. 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?
–6x + y = negative 6 x plus y equals StartFraction 3 over 2 EndFraction.
6x + y = 6 x plus y equals StartFraction 3 over 2 EndFraction.
–6x + 2y = negative 6 x plus 2 y equals StartFraction 3 over 2 EndFraction.
6x + 2y = 6 x plus 2 y equals StartFraction 3 over 2 EndFraction.
Tomas wrote the equation y = 3x +y equals 3 x plus StartFraction 3 over 4 EndFraction.. 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?
–6x + y = negative 6 x plus y equals StartFraction 3 over 2 EndFraction.
6x + y = 6 x plus y equals StartFraction 3 over 2 EndFraction.
–6x + 2y = negative 6 x plus 2 y equals StartFraction 3 over 2 EndFraction.
6x + 2y = 6 x plus 2 y equals StartFraction 3 over 2 EndFraction.
Answers
GPT-4o mini
To determine which of Sandra's equations has the same solutions as Tomas's equation \(y = 3x + \frac{3}{4}\), we can manipulate the original equation into various forms.
Starting with Tomas's equation:
\[
y = 3x + \frac{3}{4}
\]
We can rearrange this into a standard linear equation form (Ax + By = C):
\[
y - 3x = \frac{3}{4}
\]
Multiplying every term by 4 to eliminate the fraction:
\[
4y - 12x = 3
\]
We can rearrange this into:
\[
-12x + 4y = 3
\]
Now, we can simplify this equation further by dividing by the greatest common divisor, which is 4:
\[
-3x + y = \frac{3}{4}
\]
Next, to determine Sandra's equation options, we can multiply the entire equation by -2 to see if we can match any of the provided options, since multiplying an equation by a non-zero constant does not change its solutions.
Multiplying by -2 gives:
\[
6x - 2y = -\frac{3}{2}
\]
Rearranging this, we get:
\[
6x + 2y = \frac{3}{2}
\]
Now we should compare this with the options provided:
1. \(-6x + y = -\frac{3}{2}\) (not equivalent)
2. \(6x + y = \frac{3}{2}\) (not equivalent)
3. \(-6x + 2y = -\frac{3}{2}\) (not equivalent)
4. \(6x + 2y = \frac{3}{2}\) (this matches our derived equation)
So, the equation that Sandra could have written that has all the same solutions as Tomas's equation is:
\[
\boxed{6x + 2y = \frac{3}{2}}
\]
Starting with Tomas's equation:
\[
y = 3x + \frac{3}{4}
\]
We can rearrange this into a standard linear equation form (Ax + By = C):
\[
y - 3x = \frac{3}{4}
\]
Multiplying every term by 4 to eliminate the fraction:
\[
4y - 12x = 3
\]
We can rearrange this into:
\[
-12x + 4y = 3
\]
Now, we can simplify this equation further by dividing by the greatest common divisor, which is 4:
\[
-3x + y = \frac{3}{4}
\]
Next, to determine Sandra's equation options, we can multiply the entire equation by -2 to see if we can match any of the provided options, since multiplying an equation by a non-zero constant does not change its solutions.
Multiplying by -2 gives:
\[
6x - 2y = -\frac{3}{2}
\]
Rearranging this, we get:
\[
6x + 2y = \frac{3}{2}
\]
Now we should compare this with the options provided:
1. \(-6x + y = -\frac{3}{2}\) (not equivalent)
2. \(6x + y = \frac{3}{2}\) (not equivalent)
3. \(-6x + 2y = -\frac{3}{2}\) (not equivalent)
4. \(6x + 2y = \frac{3}{2}\) (this matches our derived equation)
So, the equation that Sandra could have written that has all the same solutions as Tomas's equation is:
\[
\boxed{6x + 2y = \frac{3}{2}}
\]