To solve the system of equations:
[tex]\[
\begin{cases}
x + 2y = -8 \\
3y - x = -3
\end{cases}
\][/tex]
we will use the method of substitution or elimination to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Let's do it step by step:
### Step 1: Rewrite the equations
First, let's write down the original equations clearly:
[tex]\[ \text{(1)} \quad x + 2y = -8 \][/tex]
[tex]\[ \text{(2)} \quad 3y - x = -3 \][/tex]
### Step 2: Solve one of the equations for one variable
We can solve Equation (1) for [tex]\( x \)[/tex]:
[tex]\[ x = -8 - 2y \][/tex]
### Step 3: Substitute this expression into the other equation
Substitute [tex]\( x = -8 - 2y \)[/tex] into Equation (2):
[tex]\[ 3y - (-8 - 2y) = -3 \][/tex]
Simplify the equation:
[tex]\[ 3y + 8 + 2y = -3 \][/tex]
[tex]\[ 5y + 8 = -3 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[ 5y = -3 - 8 \][/tex]
[tex]\[ 5y = -11 \][/tex]
[tex]\[ y = -\frac{11}{5} \][/tex]
### Step 5: Substitute [tex]\( y \)[/tex] back into one of the original equations to find [tex]\( x \)[/tex]
Substitute [tex]\( y = -\frac{11}{5} \)[/tex] back into the expression we found for [tex]\( x \)[/tex]:
[tex]\[ x = -8 - 2\left(-\frac{11}{5}\right) \][/tex]
[tex]\[ x = -8 + \frac{22}{5} \][/tex]
Convert [tex]\(-8\)[/tex] to a fraction with a common denominator:
[tex]\[ x = -\frac{40}{5} + \frac{22}{5} \][/tex]
[tex]\[ x = -\frac{18}{5} \][/tex]
### Step 6: Write the solution as a pair
Thus, the solution to the system of equations is:
[tex]\[ x = -\frac{18}{5} \][/tex]
[tex]\[ y = -\frac{11}{5} \][/tex]
So, the solution in the form of an ordered pair [tex]\((x, y) \)[/tex] is:
[tex]\[ \left(-\frac{18}{5}, -\frac{11}{5}\right) \][/tex]
These values satisfy both of the original equations.
