Question

unit 3 study guide: coordinate geometry and systems of linear equations and inequalities
all questions must be attempted by. the study guide must be completed and in class on that day to receive full credit. incomplete study guides and late study guides will receive a 50 for a grade. there will not be an opportunity to improve this grade.
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1. solve the following systems of equations using either the elimination or substitution method.

a. \\( \

$$\begin{cases} 9x + 4y = 32 \\\\ 3x - 4y = 20 \\end{cases}$$

\\) solution:
b. \\( \

$$\begin{cases} 5x + 4y = 20 \\\\ 5x + 3y = 35 \\end{cases}$$

\\) solution:
c. \\( \

$$\begin{cases} -3x + 4y = 12 \\\\ y = x + 5 \\end{cases}$$

\\) solution:
d. \\( \

$$\begin{cases} 4x - 2y = 10 \\\\ y = 3x - 8 \\end{cases}$$

\\) solution:

1. graph the solutions to the inequality \\( 3x + 6y>18 \\)

Explanation:

a.

Step1: Add the two equations

$$\begin{align*} 9x + 4y+(3x - 4y)&=32 + 20\\ 9x+3x+4y - 4y&=52\\ 12x&=52\\ x&=\frac{52}{12}=\frac{13}{3} \end{align*}$$

Step2: Substitute \(x = \frac{13}{3}\) into \(3x-4y = 20\)

$$\begin{align*} 3\times\frac{13}{3}-4y&=20\\ 13-4y&=20\\ -4y&=20 - 13\\ -4y&=7\\ y&=-\frac{7}{4} \end{align*}$$

Step1: Subtract the first equation from the second equation

$$\begin{align*} (5x + 3y)-(5x + 4y)&=35-20\\ 5x+3y - 5x-4y&=15\\ -y&=15\\ y&=- 15 \end{align*}$$

Step2: Substitute \(y=-15\) into \(5x + 4y=20\)

$$\begin{align*} 5x+4\times(-15)&=20\\ 5x-60&=20\\ 5x&=20 + 60\\ 5x&=80\\ x&=16 \end{align*}$$

Step1: Substitute \(y=x + 5\) into \(-3x + 4y=12\)

$$\begin{align*} -3x+4(x + 5)&=12\\ -3x+4x+20&=12\\ x+20&=12\\ x&=12-20\\ x&=-8 \end{align*}$$

Step2: Substitute \(x = - 8\) into \(y=x + 5\)

\(y=-8 + 5=-3\)

Answer:

\(x=\frac{13}{3},y =-\frac{7}{4}\)

b.