Let's solve the given system of linear equations using the method of substitution or elimination. Here are the equations:
[tex]\[ \begin{aligned}
5x - 3y &= 22 \quad \quad (1) \\
x - y &= 4 \quad \quad \quad (2)
\end{aligned} \][/tex]
### Step-by-Step Solution
#### Step 1: Solving one of the equations for one variable
Let's solve Equation (2) for [tex]\( x \)[/tex]:
[tex]\[ x - y = 4 \][/tex]
[tex]\[ x = y + 4 \quad \quad \quad (3) \][/tex]
#### Step 2: Substituting into the other equation
Now substitute Equation (3) into Equation (1):
[tex]\[ 5(y + 4) - 3y = 22 \][/tex]
#### Step 3: Simplifying the substituted equation
Expand and simplify:
[tex]\[ 5y + 20 - 3y = 22 \][/tex]
[tex]\[ 2y + 20 = 22 \][/tex]
[tex]\[ 2y = 22 - 20 \][/tex]
[tex]\[ 2y = 2 \][/tex]
Divide both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 1 \][/tex]
#### Step 4: Finding the value of [tex]\( x \)[/tex]
Now that we have [tex]\( y = 1 \)[/tex], substitute this value back into Equation (3) to find [tex]\( x \)[/tex]:
[tex]\[ x = 1 + 4 \][/tex]
[tex]\[ x = 5 \][/tex]
### Solution
The solution to the system of equations is:
[tex]\[ x = 5 \][/tex]
[tex]\[ y = 1 \][/tex]
Hence, the values that satisfy both equations are [tex]\( x = 5 \)[/tex] and [tex]\( y = 1 \)[/tex].
