To solve the system of equations:
- \( 2x + 5y = 55 \) (Equation 1)
- \( 9x - 2y = 3 \) (Equation 2)
We can use the substitution or elimination method. I will use the substitution method.
Step 1: Solve Equation 1 for \( y \)
From Equation 1:
\[ 5y = 55 - 2x \]
\[ y = \frac{55 - 2x}{5} \]
Step 2: Substitute \( y \) in Equation 2
Now, substitute the expression for \( y \) into Equation 2:
\[ 9x - 2\left(\frac{55 - 2x}{5}\right) = 3 \]
Step 3: Simplify the equation
Multiply both sides by 5 to eliminate the fraction:
\[ 5(9x) - 2(55 - 2x) = 15 \]
\[ 45x - 110 + 4x = 15 \]
Combine like terms:
\[ 49x - 110 = 15 \]
Step 4: Solve for \( x \)
Add 110 to both sides:
\[ 49x = 125 \]
Now, divide by 49:
\[ x = \frac{125}{49} \approx 2.55 \text{ (rounded to nearest hundredth)} \]
Step 5: Substitute \( x \) back to find \( y \)
Now substitute \( x \) back into the equation for \( y \):
\[ y = \frac{55 - 2\left(\frac{125}{49}\right)}{5} \]
Calculate \( 2 \times \frac{125}{49} \):
\[ 2 \times \frac{125}{49} = \frac{250}{49} \]
Now substitute back into the equation for \( y \):
\[ y = \frac{55 - \frac{250}{49}}{5} \]
Converting 55 to a fraction:
\[ 55 = \frac{2695}{49} \]
Now we have:
\[ y = \frac{\frac{2695 - 250}{49}}{5} \]
Calculate \( 2695 - 250 = 2445 \):
\[ y = \frac{\frac{2445}{49}}{5} = \frac{2445}{245} \]
Now we divide:
\[ y = \frac{2445 \div 245}{1} \approx 9.98 \text{ (rounded to nearest hundredth)} \]
Final Result
The solution to the system of equations is approximately:
\[ x \approx 2.55, \quad y \approx 9.98 \]