Question

1. algebraically determine the value of x when $f(x) = g(x)$ in the system below; $f(x) = 15x + 30$ $g(x) = 20x - 5$ lo3: solve each system using elimination. 8. $-10x + 2y = -32$ $-2x - 2y = -16$ 9. $9x - 8y = -21$ $-9x + 3y = 36$ 10. $3x - 8y = 19$ $3x + 5y = -46$

Explanation:

Problem 7

Step1: Set \( f(x) = g(x) \)

Since \( f(x) = 15x + 30 \) and \( g(x) = 20x - 5 \), we set them equal: \( 15x + 30 = 20x - 5 \)

Step2: Subtract \( 15x \) from both sides

\( 30 = 5x - 5 \)

Step3: Add 5 to both sides

\( 35 = 5x \)

Step4: Divide by 5

\( x = \frac{35}{5} = 7 \)

Step1: Add the two equations

We have \( -10x + 2y = -32 \) and \( -2x - 2y = -16 \). Adding them: \( (-10x - 2x) + (2y - 2y) = -32 - 16 \)

Step2: Simplify

\( -12x = -48 \)

Step3: Solve for \( x \)

\( x = \frac{-48}{-12} = 4 \)

Step4: Substitute \( x = 4 \) into one equation (e.g., \( -2x - 2y = -16 \))

\( -2(4) - 2y = -16 \)
\( -8 - 2y = -16 \)

Step5: Solve for \( y \)

Add 8 to both sides: \( -2y = -8 \)
\( y = \frac{-8}{-2} = 4 \)

Step1: Add the two equations

We have \( 9x - 8y = -21 \) and \( -9x + 3y = 36 \). Adding them: \( (9x - 9x) + (-8y + 3y) = -21 + 36 \)

Step2: Simplify

\( -5y = 15 \)

Step3: Solve for \( y \)

\( y = \frac{15}{-5} = -3 \)

Step4: Substitute \( y = -3 \) into one equation (e.g., \( 9x - 8y = -21 \))

\( 9x - 8(-3) = -21 \)
\( 9x + 24 = -21 \)

Step5: Solve for \( x \)

Subtract 24: \( 9x = -45 \)
\( x = \frac{-45}{9} = -5 \)

Answer:

\( x = 7 \)

Problem 8