Solve the following paris of linear (simultaneous) equation using method of elimination by substution:
0.2x + 0.1y = 25
2(x - 2) - 1.6y = 116

Answer

To solve the given pair of linear equations using the method of elimination by substitution, we will follow these steps:

### Step 1: Write down the equations
The equations given are:
1. \( 0.2x + 0.1y = 25 \) (Equation 1)
2. \( 2(x - 2) - 1.6y = 116 \) (Equation 2)

### Step 2: Simplify Equation 2
First, let's simplify Equation 2:
\[ 2(x - 2) - 1.6y = 116 \]
Distributing the 2:
\[ 2x - 4 - 1.6y = 116 \]
Adding 4 to both sides:
\[ 2x - 1.6y = 120 \quad (Equation 2') \]

### Step 3: Solve for one variable
Now, we will solve Equation 2' for \( x \):
\[ 2x = 120 + 1.6y \]
Dividing by 2:
\[ x = 60 + 0.8y \quad (Equation 3) \]

### Step 4: Substitute x in Equation 1
Now we substitute \( x \) from Equation 3 into Equation 1:
\[ 0.2(60 + 0.8y) + 0.1y = 25 \]
Distributing \( 0.2 \):
\[ 12 + 0.16y + 0.1y = 25 \]
Combining like terms:
\[ 12 + 0.26y = 25 \]
Subtracting 12 from both sides:
\[ 0.26y = 13 \]
Dividing by 0.26:
\[ y = \frac{13}{0.26} = 50 \]

### Step 5: Find the value of x
Now that we have \( y \), we can find \( x \) using Equation 3:
\[ x = 60 + 0.8(50) \]
Calculating:
\[ x = 60 + 40 = 100 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = 100, \quad y = 50 \]

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