Question

tell whether each relationship is a direct variation. if so, identify the constant of variation.
8.

x	-1	2	5	9
y	4	7	10	14

9.

x	-2	2	6	10
y	1	-1	-3	-5

1. write the equation $2x - 2y = 4$ in slope - intercept form, and then graph.

Explanation:

Problem 8:

Step1: Recall direct variation formula

A direct variation relationship is of the form \( y = kx \), where \( k \) is a constant (the constant of variation). This means that \( \frac{y}{x} \) should be constant for all pairs of \( x \) and \( y \) (where \( x
eq0 \)).

Step2: Calculate \( \frac{y}{x} \) for each pair

• For \( x = -1, y = 4 \): \( \frac{y}{x}=\frac{4}{-1}=-4 \)
• For \( x = 2, y = 7 \): \( \frac{y}{x}=\frac{7}{2} = 3.5 \)
• For \( x = 5, y = 10 \): \( \frac{y}{x}=\frac{10}{5}=2 \)
• For \( x = 9, y = 14 \): \( \frac{y}{x}=\frac{14}{9}\approx1.56 \)

Since the ratios \( \frac{y}{x} \) are not constant, the relationship is not a direct variation.

Step1: Recall direct variation formula

A direct variation relationship is \( y = kx \), so \( \frac{y}{x} \) should be constant for all \( x
eq0 \).

Step2: Calculate \( \frac{y}{x} \) for each pair

• For \( x = -2, y = 1 \): \( \frac{y}{x}=\frac{1}{-2}=-0.5 \)
• For \( x = 2, y = -1 \): \( \frac{y}{x}=\frac{-1}{2}=-0.5 \)
• For \( x = 6, y = -3 \): \( \frac{y}{x}=\frac{-3}{6}=-0.5 \)
• For \( x = 10, y = -5 \): \( \frac{y}{x}=\frac{-5}{10}=-0.5 \)

Since \( \frac{y}{x}=-0.5 \) (constant), the relationship is a direct variation with \( k = -0.5 \).

Step1: Recall slope - intercept form

The slope - intercept form of a linear equation is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. We need to solve the equation \( 2x-2y = 4 \) for \( y \).

Step2: Isolate \( y \)

• Start with \( 2x-2y = 4 \)
• Subtract \( 2x \) from both sides: \( - 2y=-2x + 4 \)
• Divide each term by - 2: \( y=\frac{-2x}{-2}+\frac{4}{-2} \)
• Simplify: \( y=x - 2 \)

To graph \( y=x - 2 \):

• The y - intercept \( b=-2 \), so the line passes through the point \( (0,-2) \).
• The slope \( m = 1 \), which means for every 1 unit we move to the right (increase \( x \) by 1), we move up 1 unit (increase \( y \) by 1). So from \( (0,-2) \), we can find another point: \( (0 + 1,-2 + 1)=(1,-1) \), \( (1+1,-1 + 1)=(2,0) \), etc. Then draw a line through these points.

Answer:

Not a direct variation

Problem 9: