Answer :

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].