The problem asks us to solve a system of linear equations using the elimination method. I will solve problem number 12. The system of equations is: $3x + 2y = 4$ $4x + 5y = 10$

1. Problem Description

The problem asks us to solve a system of linear equations using the elimination method. I will solve problem number

1

2. The system of equations is:

$3x + 2y = 4$

$4x + 5y = 10$

2. Solution Steps

We aim to eliminate one of the variables by multiplying the equations by appropriate constants so that the coefficients of one variable are equal in magnitude but opposite in sign.

Step 1: Multiply the first equation by 4 and the second equation by -

3. This will make the coefficients of x be 12 and -12 respectively.

$4(3x + 2y) = 4(4)$ => $12x + 8y = 16$

$-3(4x + 5y) = -3(10)$ => $-12x - 15y = -30$

Step 2: Add the two new equations.

$(12x + 8y) + (-12x - 15y) = 16 + (-30)$

$12x - 12x + 8y - 15y = -14$

$-7y = -14$

Step 3: Solve for y.

$y = \frac{-14}{-7}$

$y = 2$

Step 4: Substitute the value of y back into one of the original equations to solve for x. We can use the first equation:

$3x + 2y = 4$

$3x + 2(2) = 4$

$3x + 4 = 4$

$3x = 4 - 4$

$3x = 0$

$x = \frac{0}{3}$

$x = 0$

3. Final Answer

$x = 0$ and $y = 2$