To solve the system of equations:
1. [tex]\(-2x - \frac{2y}{4} = 12\)[/tex]
2. [tex]\(\frac{x}{5} - \frac{y}{3} = 5\)[/tex]
We'll start by simplifying the equations.
### Simplifying the First Equation:
The first equation is:
[tex]\[
-2x - \frac{2y}{4} = 12
\][/tex]
Simplify [tex]\(\frac{2y}{4}\)[/tex] to [tex]\(\frac{y}{2}\)[/tex]:
[tex]\[
-2x - \frac{y}{2} = 12
\][/tex]
To eliminate the fraction, multiply the entire equation by 2:
[tex]\[
-4x - y = 24 \quad \Rightarrow \quad 4x + y = -24 \quad \text{(Equation 3)}
\][/tex]
### Simplifying the Second Equation:
The second equation is:
[tex]\[
\frac{x}{5} - \frac{y}{3} = 5
\][/tex]
To eliminate the fractions, multiply the entire equation by 15 (which is the least common multiple of 5 and 3):
[tex]\[
15 \left(\frac{x}{5}\right) - 15 \left(\frac{y}{3}\right) = 15 \times 5 \\
3x - 5y = 75 \quad \text{(Equation 4)}
\][/tex]
Now, we have the simplified system of equations:
1. [tex]\(4x + y = -24\)[/tex]
2. [tex]\(3x - 5y = 75\)[/tex]
### Solving the System of Equations:
We can use the method of substitution or elimination. Here, we'll use elimination.
Step 1: Align the equations:
[tex]\[
4x + y = -24 \quad \text{(Equation 3)}
\][/tex]
[tex]\[
3x - 5y = 75 \quad \text{(Equation 4)}
\][/tex]
Step 2: Make the coefficients of y in both equations equal by multiplying the first equation by 5:
[tex]\[
5(4x + y) = 5(-24) \\
20x + 5y = -120 \quad \text{(Equation 5)}
\][/tex]
Step 3: Add Equation 5 and Equation 4 to eliminate y:
[tex]\[
(20x + 5y) + (3x - 5y) = -120 + 75 \\
20x + 3x + 5y - 5y = -120 + 75 \\
23x = -45 \\
x = \frac{-45}{23} \\
x = -\frac{45}{23}
\][/tex]
Step 4: Substitute [tex]\(x\)[/tex] back into Equation 3 to find [tex]\(y\)[/tex]:
[tex]\[
4 \left( -\frac{45}{23} \right) + y = -24 \\
-\frac{180}{23} + y = -24
\][/tex]
First, rewrite [tex]\(-24\)[/tex] as a fraction:
[tex]\[
-24 = -\frac{552}{23}
\][/tex]
So:
[tex]\[
-\frac{180}{23} + y = -\frac{552}{23} \\
y = -\frac{552}{23} + \frac{180}{23} \\
y = -\frac{552 - 180}{23} \\
y = -\frac{372}{23}
\][/tex]
Thus:
[tex]\[
y = -\frac{372}{23}
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = -\frac{45}{23}, \quad y = -\frac{372}{23}
\][/tex]
