Questions: Solve the system of equations by the substitution method. 4x + 3y = 27 y = -(4/3)x + 9 (x, y) = ( )

Transcript text: Solve the system of equations by the substitution method. \[ \begin{array}{l} \left\{\begin{array}{r} 4 x+3 y=27 \\ y=-\frac{4}{3} x+9 \end{array}\right. \\ (x, y)=(\square) \end{array} \]

Solution

Solution Steps

To solve the system of equations using the substitution method, follow these steps:

Substitute the expression for \( y \) from the second equation into the first equation.
Solve the resulting equation for \( x \).
Substitute the value of \( x \) back into the second equation to find \( y \).

Step 1: Substitute the Expression for \( y \)

Given the system of equations: \[ \begin{cases} 4x + 3y = 27 \\ y = -\frac{4}{3}x + 9 \end{cases} \]

We will use the substitution method. First, substitute the expression for \( y \) from the second equation into the first equation.

Step 2: Substitute and Simplify

Substitute \( y = -\frac{4}{3}x + 9 \) into \( 4x + 3y = 27 \):

\[ 4x + 3\left(-\frac{4}{3}x + 9\right) = 27 \]

Simplify inside the parentheses:

\[ 4x + 3 \left(-\frac{4}{3}x\right) + 3 \cdot 9 = 27 \]

\[ 4x - 4x + 27 = 27 \]

Step 3: Solve for \( x \)

Combine like terms:

\[ 0x + 27 = 27 \]

This simplifies to:

\[ 27 = 27 \]

This is a true statement, which means the system of equations is dependent, and there are infinitely many solutions. The equations represent the same line.