For solving each pair of equations, in this exercise, use the method of elimination by equating coefficients :
13 + 2y = 9x
3y = 7x

Answer

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

### Step 1: Write down the equations
The given equations are:
1. \( 13 + 2y = 9x \) (Equation 1)
2. \( 3y = 7x \) (Equation 2)

### Step 2: Rearrange the equations
We can rearrange both equations to express them in standard form (Ax + By = C):
1. From Equation 1: \( 9x - 2y = 13 \)
2. From Equation 2: \( 7x - 3y = 0 \)

### Step 3: Multiply the equations to equalize coefficients
To eliminate one of the variables, we can multiply the equations to make the coefficients of \( y \) equal. We will multiply Equation 1 by 3 and Equation 2 by 2:
1. \( 3(9x - 2y) = 3(13) \) → \( 27x - 6y = 39 \) (Equation 3)
2. \( 2(7x - 3y) = 2(0) \) → \( 14x - 6y = 0 \) (Equation 4)

### Step 4: Subtract the equations
Now, we can subtract Equation 4 from Equation 3:
\[ (27x - 6y) - (14x - 6y) = 39 - 0 \]
This simplifies to:
\[ 27x - 14x = 39 \]
\[ 13x = 39 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 13:
\[ x = \frac{39}{13} = 3 \]

### Step 6: Substitute \( x \) back to find \( y \)
Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). We will use Equation 2:
\[ 3y = 7x \]
Substituting \( x = 3 \):
\[ 3y = 7(3) \]
\[ 3y = 21 \]
Now, divide both sides by 3:
\[ y = \frac{21}{3} = 7 \]

### Final Solution
The solution to the system of equations is:
\[ x = 3, \quad y = 7 \]