Let's solve the system of linear equations step-by-step:
### Step 1: Write down the given system of equations
[tex]\[
\begin{cases}
x + y = 55 \quad \text{(1)} \\
4x + 2y = 170 \quad \text{(2)}
\end{cases}
\][/tex]
### Step 2: Express one variable in terms of the other from one of the equations
Let's solve Equation (1) for [tex]\( y \)[/tex]:
[tex]\[
y = 55 - x \quad \text{(3)}
\][/tex]
### Step 3: Substitute this expression into the second equation
Substitute [tex]\( y = 55 - x \)[/tex] into Equation (2):
[tex]\[
4x + 2(55 - x) = 170
\][/tex]
### Step 4: Simplify and solve for [tex]\( x \)[/tex]
[tex]\[
4x + 110 - 2x = 170
\][/tex]
Combine like terms:
[tex]\[
2x + 110 = 170
\][/tex]
Subtract 110 from both sides:
[tex]\[
2x = 60
\][/tex]
Divide both sides by 2:
[tex]\[
x = 30
\][/tex]
### Step 5: Substitute [tex]\( x = 30 \)[/tex] back into Equation (3) to solve for [tex]\( y \)[/tex]
[tex]\[
y = 55 - 30
\][/tex]
Simplify:
[tex]\[
y = 25
\][/tex]
### Step 6: Display the solution
The solution to the system of equations is:
[tex]\[
x = 30, \quad y = 25
\][/tex]
To verify, substitute [tex]\( x = 30 \)[/tex] and [tex]\( y = 25 \)[/tex] back into the original equations:
1. [tex]\( x + y = 30 + 25 = 55 \)[/tex] (satisfies Equation (1))
2. [tex]\( 4x + 2y = 4(30) + 2(25) = 120 + 50 = 170 \)[/tex] (satisfies Equation (2))
Thus, the solution is correct. The values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are [tex]\( x = 30 \)[/tex] and [tex]\( y = 25 \)[/tex].
