Let's analyze Tomas's equation and compare it to each of Sandra's given equations to determine which of Sandra's equations has all the same solutions as Tomas's equation.
Tomas's equation is given as:
[tex]\[ y = 3x + \frac{3}{4} \][/tex]
We need to transform each of Sandra's given equations into the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
1. The first equation:
[tex]\[ -6x + y = \frac{3}{2} \][/tex]
We can solve for [tex]\( y \)[/tex]:
[tex]\[ y = 6x + \frac{3}{2} \][/tex]
This equation has a slope of 6, which is not equal to Tomas's slope of 3.
2. The second equation:
[tex]\[ 6x + y = \frac{3}{2} \][/tex]
We can solve for [tex]\( y \)[/tex]:
[tex]\[ y = -6x + \frac{3}{2} \][/tex]
This equation has a slope of -6, which is not equal to Tomas's slope of 3.
3. The third equation:
[tex]\[ -6x + 2y = \frac{3}{2} \][/tex]
We can solve for [tex]\( y \)[/tex]:
First, isolate [tex]\( y \)[/tex] by dividing everything by 2:
[tex]\[ 2y = 6x + \frac{3}{2} \][/tex]
[tex]\[ y = 3x + \frac{3}{4} \][/tex]
This equation simplifies exactly to Tomas's equation.
4. The fourth equation:
[tex]\[ 6x + 2y = \frac{3}{2} \][/tex]
We can solve for [tex]\( y \)[/tex]:
First, isolate [tex]\( y \)[/tex] by dividing everything by 2:
[tex]\[ 2y = -6x + \frac{3}{2} \][/tex]
[tex]\[ y = -3x + \frac{3}{4} \][/tex]
This equation has a slope of -3, which is not equal to Tomas's slope of 3.
Among all four of Sandra's equations, only the third equation:
[tex]\[ -6x + 2y = \frac{3}{2} \][/tex]
is equivalent to Tomas's equation:
[tex]\[ y = 3x + \frac{3}{4} \][/tex]
Thus, Sandra's equation which has all the same solutions as Tomas's equation is:
[tex]\[ -6x + 2y = \frac{3}{2} \][/tex]
The correct choice is:
[tex]\[ \boxed{-6x + 2y = \frac{3}{2}} \][/tex]
