Question

solve using elimination.
5x - 6y = 9
3x - 2y = 15
(\boxed{ }, \boxed{ }))

Explanation:

To solve the system of equations using the elimination method, we start with the given equations:

$$
LATEXBLOCK0
$$

Step 1: Make the coefficients of \( y \) opposites

Notice that if we multiply equation (2) by \( -3 \), the coefficient of \( y \) will become \( 6 \), which is the opposite of \( -6 \) in equation (1).

Multiply equation (2) by \( -3 \):

$$ -3(3x - 2y) = -3(15) $$

Simplify:

$$ -9x + 6y = -45 \quad \text{(3)} $$

Step 2: Add the equations to eliminate \( y \)

Now, add equation (1) and equation (3):

$$
LATEXBLOCK1
$$

Step 3: Solve for \( x \)

Divide both sides by \( -4 \):

$$ x = \frac{-36}{-4} = 9 $$

Step 4: Substitute \( x = 9 \) back into one of the original equations to solve for \( y \)

We'll use equation (2): \( 3x - 2y = 15 \)

Substitute \( x = 9 \):

$$ 3(9) - 2y = 15 $$

Simplify:

$$ 27 - 2y = 15 $$

Subtract \( 27 \) from both sides:

$$ -2y = 15 - 27 \\ -2y = -12 $$

Divide both sides by \( -2 \):

$$ y = \frac{-12}{-2} = 6 $$

Final Solution

The solution to the system is \( \boldsymbol{(9, 6)} \).

Answer:

To solve the system of equations using the elimination method, we start with the given equations:

$$
LATEXBLOCK0
$$

Step 1: Make the coefficients of \( y \) opposites

Notice that if we multiply equation (2) by \( -3 \), the coefficient of \( y \) will become \( 6 \), which is the opposite of \( -6 \) in equation (1).

Multiply equation (2) by \( -3 \):

$$ -3(3x - 2y) = -3(15) $$

Simplify:

$$ -9x + 6y = -45 \quad \text{(3)} $$

Step 2: Add the equations to eliminate \( y \)

Now, add equation (1) and equation (3):

$$
LATEXBLOCK1
$$

Step 3: Solve for \( x \)

Divide both sides by \( -4 \):

$$ x = \frac{-36}{-4} = 9 $$

Step 4: Substitute \( x = 9 \) back into one of the original equations to solve for \( y \)

We'll use equation (2): \( 3x - 2y = 15 \)

Substitute \( x = 9 \):

$$ 3(9) - 2y = 15 $$

Simplify:

$$ 27 - 2y = 15 $$

Subtract \( 27 \) from both sides:

$$ -2y = 15 - 27 \\ -2y = -12 $$

Divide both sides by \( -2 \):

$$ y = \frac{-12}{-2} = 6 $$

Final Solution

The solution to the system is \( \boldsymbol{(9, 6)} \).