To solve the system of equations given:
[tex]\[ -\frac{2}{3} y = -\frac{6}{3} + \frac{2}{3} x \][/tex]
[tex]\[ 10 x = -2 y - 2 \][/tex]
we can follow a systematic procedure step-by-step.
### Step 1: Simplify each equation
First, simplify the first equation.
[tex]\[ -\frac{2}{3} y = -2 + \frac{2}{3} x \][/tex]
Next, simplify the second equation.
[tex]\[ 10 x = -2 y - 2 \][/tex]
### Step 2: Solve for one variable in terms of the other
Let's solve the first equation for [tex]\(y\)[/tex].
Rearrange the first equation to isolate [tex]\(y\)[/tex]:
[tex]\[ -\frac{2}{3} y = -2 + \frac{2}{3} x \][/tex]
Multiply through by [tex]\(-3\)[/tex] to clear the fraction:
[tex]\[ 2y = 6 - 2x \][/tex]
[tex]\[ y = 3 - x \][/tex]
Now we have:
[tex]\[ y = 3 - x \][/tex]
### Step 3: Substitute into the other equation
Substitute [tex]\( y = 3 - x \)[/tex] into the second equation:
[tex]\[ 10 x = -2 y - 2 \][/tex]
[tex]\[ 10 x = -2 (3 - x) - 2 \][/tex]
Distribute the [tex]\(-2\)[/tex]:
[tex]\[ 10 x = -6 + 2x - 2 \][/tex]
Combine like terms:
[tex]\[ 10 x = 2 x - 8 \][/tex]
[tex]\[ 10 x - 2 x = -8 \][/tex]
[tex]\[ 8 x = -8 \][/tex]
[tex]\[ x = -1 \][/tex]
### Step 4: Solve for the other variable
Substitute [tex]\( x = -1 \)[/tex] back into [tex]\( y = 3 - x \)[/tex]:
[tex]\[ y = 3 - (-1) \][/tex]
[tex]\[ y = 3 + 1 \][/tex]
[tex]\[ y = 4 \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ x = -1 \][/tex]
[tex]\[ y = 4 \][/tex]
Therefore, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations are [tex]\((-1, 4)\)[/tex].
