To determine which of Sandra's equations matches Tomas's equation [tex]\( y = 3x + \frac{3}{4} \)[/tex] for all values of [tex]\( x \)[/tex], we need to transform each of Sandra's provided equations into a similar form and see which one matches Tomas's equation.
1. Equation: [tex]\( -6x + y = \frac{3}{2} \)[/tex]
- To solve for [tex]\( y \)[/tex], isolate [tex]\( y \)[/tex]:
```
y = 6x + \frac{3}{2}
```
- This form of [tex]\( y \)[/tex] does not match [tex]\( y = 3x + \frac{3}{4} \)[/tex].
2. Equation: [tex]\( 6x + y = \frac{3}{2} \)[/tex]
- To solve for [tex]\( y \)[/tex], isolate [tex]\( y \)[/tex]:
```
y = -6x + \frac{3}{2}
```
- This form of [tex]\( y \)[/tex] does not match [tex]\( y = 3x + \frac{3}{4} \)[/tex].
3. Equation: [tex]\( -6x + 2y = \frac{3}{2} \)[/tex]
- To solve for [tex]\( y \)[/tex], isolate [tex]\( y \)[/tex]:
```
2y = 6x + \frac{3}{2}
y = 3x + \frac{3}{4}
```
- This form of [tex]\( y \)[/tex] matches [tex]\( y = 3x + \frac{3}{4} \)[/tex].
4. Equation: [tex]\( 6x + 2y = \frac{3}{2} \)[/tex]
- To solve for [tex]\( y \)[/tex], isolate [tex]\( y \)[/tex]:
```
2y = -6x + \frac{3}{2}
y = -3x + \frac{3}{4}
```
- This form of [tex]\( y \)[/tex] does not match [tex]\( y = 3x + \frac{3}{4} \)[/tex].
Thus, the only equation that correctly transforms to [tex]\( y = 3x + \frac{3}{4} \)[/tex] and therefore has all the same solutions as Tomas's equation is:
[tex]\[ -6x + 2y = \frac{3}{2} \][/tex]
So, Sandra's equation could be:
[tex]\[ -6x + 2y = \frac{3}{2} \][/tex]
