Question

Solve the following pair of equations using the cross-multiplication method: \[ 2x + 3y - 46 = 0 \] \[ 3x + 5y - 74 = 0 \]

Hint:

In the cross-multiplication method, solve for \( x \) and \( y \) using the formulas that involve the coefficients and constants of the system of equations.

Answer 1

We are given the pair of equations: 1. \( 2x + 3y = 46 \) 2. \( 3x + 5y = 74 \) Step 1: Write the equations in standard form. \[ 2x + 3y = 46 \quad \text{(Equation 1)} \] \[ 3x + 5y = 74 \quad \text{(Equation 2)} \] Step 2: Use the cross-multiplication method. The cross-multiplication method involves solving the system using the following formulas: \[ x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1} \quad \text{and} \quad y = \frac{a_1c_2 - a_2c_1}{a_1b_2 - a_2b_1} \] For the given system: - \( a_1 = 2, b_1 = 3, c_1 = 46 \) - \( a_2 = 3, b_2 = 5, c_2 = 74 \) Thus, the formula for \( x \) becomes: \[ x = \frac{3(74) - 5(46)}{2(5) - 3(3)} = \frac{222 - 230}{10 - 9} = \frac{-8}{1} = -8. \] The formula for \( y \) becomes: \[ y = \frac{2(74) - 3(46)}{2(5) - 3(3)} = \frac{148 - 138}{10 - 9} = \frac{10}{1} = 10. \]
Conclusion:
The solution to the system of equations is \( x = -8 \) and \( y = 10 \).