Question

1. the table shows values of y as a function of x. write the equation that describes the linear relationship shown in the table of values.

x	45	60	75
y	8	9	10

y=1/15x+5

1. what is the slope to the line 6x - 3y = 12?

17.
a. what is the slope of a line parallel to the given line?
b. give a possible equation of a line parallel to the given line.

Explanation:

Question 15

Step1: Recall linear equation form

The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. To find \( m \), we use the formula for slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take two points from the table, say \( (x_1,y_1)=(45,8) \) and \( (x_2,y_2)=(60,9) \).
\( m=\frac{9 - 8}{60 - 45}=\frac{1}{15} \)

Step2: Find the y - intercept \( b \)

Substitute \( x = 45 \), \( y = 8 \) and \( m=\frac{1}{15} \) into \( y=mx + b \).
\( 8=\frac{1}{15}\times45 + b \)
\( 8 = 3 + b \)
Subtract 3 from both sides: \( b=8 - 3=5 \)

Step3: Write the equation

Using \( m=\frac{1}{15} \) and \( b = 5 \), the equation is \( y=\frac{1}{15}x+5 \)

Step1: Rewrite the equation in slope - intercept form

We want to rewrite \( 6x-3y = 12 \) in the form \( y=mx + b \). First, isolate \( y \).
Subtract \( 6x \) from both sides: \( - 3y=-6x + 12 \)
Divide each term by - 3: \( y=\frac{-6x}{-3}+\frac{12}{-3} \)
\( y = 2x-4 \)

Step2: Identify the slope

In the slope - intercept form \( y = mx + b \), the coefficient of \( x \) is the slope. So for \( y = 2x-4 \), \( m = 2 \)

Step1: Calculate the slope of the given line

Using the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \), with \( (x_1,y_1)=(0,-2) \) and \( (x_2,y_2)=(3,-4) \)
\( m=\frac{-4-(-2)}{3 - 0}=\frac{-4 + 2}{3}=\frac{-2}{3} \)

Step2: Slope of parallel line

Parallel lines have the same slope. So the slope of a line parallel to the given line is also \( -\frac{2}{3} \)

Answer:

\( y=\frac{1}{15}x + 5 \)

Question 16