Solve the following paris of linear (simultaneous) equation using method of elimination by substution:
3x + 2y = 11
2x - 3y + 10 =0

Answer

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

### Step 1: Write the equations
The given equations are:
1. \( 3x + 2y = 11 \) (Equation 1)
2. \( 2x - 3y + 10 = 0 \) (Equation 2)

### Step 2: Rearrange Equation 2 to express \( x \) in terms of \( y \)
From Equation 2, we can rearrange it to isolate \( x \):
\[ 2x - 3y + 10 = 0 \implies 2x = 3y - 10 \implies x = \frac{3y - 10}{2} \]

### Step 3: Substitute \( x \) in Equation 1
Now, we will substitute the expression for \( x \) into Equation 1:
\[ 3\left(\frac{3y - 10}{2}\right) + 2y = 11 \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ \frac{9y - 30}{2} + 2y = 11 \]
To eliminate the fraction, multiply the entire equation by 2:
\[ 9y - 30 + 4y = 22 \]

### Step 5: Combine like terms
Combine the terms involving \( y \):
\[ 13y - 30 = 22 \]

### Step 6: Solve for \( y \)
Now, add 30 to both sides:
\[ 13y = 52 \]
Now, divide by 13:
\[ y = 4 \]

### Step 7: Substitute \( y \) back to find \( x \)
Now that we have \( y \), substitute it back into the expression for \( x \):
\[ x = \frac{3(4) - 10}{2} = \frac{12 - 10}{2} = \frac{2}{2} = 1 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = 1, \quad y = 4 \]

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