Question

use the system \\(\

$$\begin{cases} 3x - 2y = 11 \\\\ y = \\frac{1}{2}x - 1 \\end{cases}$$

\\) for questions 5 - 6.

1. describe why using substitution is an efficient method to solving the system.
2. solve the system.

Explanation:

Step1: Identify the system

The system is \(

$$\begin{cases}3x - 2y = 11\\y=\frac{1}{2}x - 1\end{cases}$$

\). Since one equation (\(y=\frac{1}{2}x - 1\)) is already solved for \(y\), substitution is efficient as we can directly substitute \(y\) into the first equation.

Step2: Substitute \(y\) into the first equation

Substitute \(y=\frac{1}{2}x - 1\) into \(3x - 2y = 11\):

$$
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$$

Step3: Solve for \(x\)

Subtract 2 from both sides: \(2x = 11 - 2 = 9\), then divide by 2: \(x=\frac{9}{2}=4.5\)

Step4: Solve for \(y\)

Substitute \(x = 4.5\) into \(y=\frac{1}{2}x - 1\): \(y=\frac{1}{2}(4.5)-1 = 2.25 - 1 = 1.25=\frac{5}{4}\)

Answer:

The solution to the system is \(x = \frac{9}{2}\), \(y=\frac{5}{4}\) (or \(x = 4.5\), \(y = 1.25\))