Solve using substitution. -3x + 2y = 9, 4x - 3y = -9

Steps to Solve

1. Solve the first equation for y

Start with the first equation:
$$ -3x + 2y = 9 $$

Add $3x$ to both sides:
$$ 2y = 3x + 9 $$

Now, divide by 2 to solve for $y$:
$$ y = \frac{3}{2}x + \frac{9}{2} $$

2. Substitute y in the second equation

Next, take the expression for $y$ and substitute it into the second equation:
$$ 4x - 3y = -9 $$

Substituting for $y$:
$$ 4x - 3\left(\frac{3}{2}x + \frac{9}{2}\right) = -9 $$

3. Simplify the second equation

Distribute $-3$ into the parentheses:
$$ 4x - \frac{9}{2}x - \frac{27}{2} = -9 $$

Combine like terms ($4x$ and $-\frac{9}{2}x$):
To combine them, convert $4x$ into halves:
$$ \frac{8}{2}x - \frac{9}{2}x = -\frac{1}{2}x $$

So now we have:
$$ -\frac{1}{2}x - \frac{27}{2} = -9 $$

4. Isolate x

Add $\frac{27}{2}$ to both sides:
$$ -\frac{1}{2}x = -9 + \frac{27}{2} $$

Convert $-9$ into halves:
$$ -9 = -\frac{18}{2} $$

Now the equation becomes:
$$ -\frac{1}{2}x = -\frac{18}{2} + \frac{27}{2} $$
$$ -\frac{1}{2}x = \frac{9}{2} $$

Multiply both sides by $-2$:
$$ x = -9 $$

5. Substitute x back to find y

Now substitute $x = -9$ back into the expression for $y$:
$$ y = \frac{3}{2}(-9) + \frac{9}{2} $$
$$ y = -\frac{27}{2} + \frac{9}{2} $$
$$ y = -\frac{18}{2} = -9 $$