To solve the given system of linear equations, we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. Let's start by writing the system of equations clearly:
1. [tex]\[\frac{1}{3} x - \frac{1}{2} y = 4\][/tex]
2. [tex]\[5x - y = -5\][/tex]
### Step-by-Step Solution
Step 1: Eliminate fractions in Equation 1
Multiply both sides of Equation 1 by 6 to eliminate the fractions:
[tex]\[6 \left(\frac{1}{3} x - \frac{1}{2} y \right) = 6 \cdot 4\][/tex]
This simplifies to:
[tex]\[ 2x - 3y = 24 \][/tex]
So, the system of equations becomes:
1. [tex]\( 2x - 3y = 24 \)[/tex]
2. [tex]\( 5x - y = -5 \)[/tex]
Step 2: Solve for one variable
Let's solve for [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex] using Equation 2:
[tex]\[ 5x - y = -5 \][/tex]
Rearrange to solve for [tex]\(y\)[/tex]:
[tex]\[ y = 5x + 5 \][/tex]
Step 3: Substitute y into Equation 1
Now substitute [tex]\( y = 5x + 5 \)[/tex] into Equation 1:
[tex]\[ 2x - 3(5x + 5) = 24 \][/tex]
Expand and simplify:
[tex]\[ 2x - 15x - 15 = 24 \][/tex]
[tex]\[ -13x - 15 = 24 \][/tex]
Add 15 to both sides:
[tex]\[ -13x = 39 \][/tex]
Divide by -13:
[tex]\[ x = -3 \][/tex]
Step 4: Solve for y using the value of x
Now that we have [tex]\( x = -3 \)[/tex], substitute this into [tex]\( y = 5x + 5 \)[/tex]:
[tex]\[ y = 5(-3) + 5 \][/tex]
[tex]\[ y = -15 + 5 \][/tex]
[tex]\[ y = -10 \][/tex]
Step 5: Write the solution
The solution to the system of equations is:
[tex]\[ x = -3 \][/tex]
[tex]\[ y = -10 \][/tex]
Therefore, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations are [tex]\( \mathbf{x = -3} \)[/tex] and [tex]\( \mathbf{y = -10} \)[/tex].
