Question

1. determine the slope and y-intercept of each.

$y = \frac{2}{9}x - 3$
$m = \underline{\hspace{5cm}} \sqrt{x}$ $b = \underline{\hspace{5cm}} \sqrt{x}$
$y = \frac{1}{3} - 2x$
$m = \underline{\hspace{5cm}} \sqrt{x}$ $b = \underline{\hspace{5cm}} \sqrt{x}$
$3x + 2y = 8$
$m = \underline{\hspace{5cm}} \sqrt{x}$ $b = \underline{\hspace{5cm}} \sqrt{x}$
$y = -11$
$m = \underline{\hspace{5cm}} \sqrt{x}$ $b = \underline{\hspace{5cm}} \sqrt{x}$

1. for 7-8, graph the following using the slope and y-intercept.

$y = \frac{1}{4}x + 1$

Explanation:

Problem 6 (First Equation: \( y = \frac{2}{9}x - 3 \))

Step1: Recall slope-intercept form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step2: Identify \( m \) and \( b \)

For \( y = \frac{2}{9}x - 3 \), comparing with \( y = mx + b \), we have \( m = \frac{2}{9} \) and \( b = -3 \).

Problem 6 (Second Equation: \( y = \frac{1}{3} - 2x \))

Step1: Rewrite in slope-intercept form

Rewrite \( y = \frac{1}{3} - 2x \) as \( y = -2x + \frac{1}{3} \).

Step2: Identify \( m \) and \( b \)

Comparing with \( y = mx + b \), we get \( m = -2 \) and \( b = \frac{1}{3} \).

Problem 6 (Third Equation: \( 3x + 2y = 8 \))

Step1: Solve for \( y \) to get slope-intercept form

Subtract \( 3x \) from both sides: \( 2y = -3x + 8 \). Then divide by 2: \( y = -\frac{3}{2}x + 4 \).

Step2: Identify \( m \) and \( b \)

Comparing with \( y = mx + b \), we have \( m = -\frac{3}{2} \) and \( b = 4 \).

Problem 6 (Fourth Equation: \( y = -11 \))

Answer:

Step1: Identify slope and y-intercept

From \( y = \frac{1}{4}x + 1 \), the slope \( m = \frac{1}{4} \) and the y-intercept \( b = 1 \).

Step2: Plot the y-intercept

Start by plotting the point \( (0, 1) \) on the y-axis (since the y-intercept is 1).

Step3: Use the slope to find another point

The slope \( \frac{1}{4} \) means "rise over run", so from \( (0, 1) \), rise 1 unit (up) and run 4 units (right) to get the point \( (4, 2) \).

Step4: Draw the line

Draw a straight line through the points \( (0, 1) \) and \( (4, 2) \).

Final Answers for Problem 6:

• For \( y = \frac{2}{9}x - 3 \): \( m = \boldsymbol{\frac{2}{9}} \), \( b = \boldsymbol{-3} \)
• For \( y = \frac{1}{3} - 2x \): \( m = \boldsymbol{-2} \), \( b = \boldsymbol{\frac{1}{3}} \)
• For \( 3x + 2y = 8 \): \( m = \boldsymbol{-\frac{3}{2}} \), \( b = \boldsymbol{4} \)
• For \( y = -11 \): \( m = \boldsymbol{0} \), \( b = \boldsymbol{-11} \)

Final Answer for Problem 7:

The graph is drawn by plotting \( (0, 1) \) and \( (4, 2) \) (or other points using the slope) and drawing a line through them. (Note: Since it's a graphing problem, the key is identifying the slope and y-intercept and using them to plot the line.)