Question

name: amelia b
period: 1
date:
unit 3 study guide: coordinate geometry and systems of linear equations and inequalities
all questions must be attempted by
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.
this page is graphing calculator inactive.

1. solve the following systems of equations using either the elimination or substitution method.

a. \\( \

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

\\) solution:
b. \\( \

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

\\) solution:
c. \\( \

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

\\) solution:
d. \\( \

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

\\) solution:

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

Explanation:

Step1: Solve part a

• Add the two equations:

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

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

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

Step2: Solve part b

• Subtract the first equation from the second equation:

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

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

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

Step3: Solve part c

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

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

• Substitute \(x = - 8\) into \(y=x + 5\), \(y=-8 + 5=-3\)

Step4: Solve part d

• Substitute \(y = 3x-8\) into \(4x-2y = 10\):

$$\begin{align*} 4x-2(3x - 8)&=10\\ 4x-6x + 16&=10\\ -2x&=10-16\\ -2x&=-6\\ x&=3 \end{align*}$$

• Substitute \(x = 3\) into \(y = 3x-8\), \(y=3\times3-8=1\)

Answer:

a. \((\frac{13}{3},-\frac{7}{4})\)
b. \((16,-15)\)
c. \((-8,-3)\)
d. \((3,1)\)