Question

1. table item complete the table if the cost y varies directly with the number x. (lesson 4)

number of snacks, x | cost ($), y
2 | 6
| 12
| 21
10 |

1. open response what is the equation of the line that passes through (0, 2) and (-3, 14) in slope intercept form? justify your reasoning.

(lesson 5)

Explanation:

Problem 8 (Table Completion, Direct Variation)

Step1: Find the constant of variation

Since \( y \) varies directly with \( x \), the relationship is \( y = kx \). Using \( x = 2 \), \( y = 6 \):
\( 6 = k(2) \) → \( k = \frac{6}{2} = 3 \). So the equation is \( y = 3x \).

Step2: Find \( x \) for \( y = 12 \)

Substitute \( y = 12 \) into \( y = 3x \):
\( 12 = 3x \) → \( x = \frac{12}{3} = 4 \).

Step3: Find \( x \) for \( y = 21 \)

Substitute \( y = 21 \) into \( y = 3x \):
\( 21 = 3x \) → \( x = \frac{21}{3} = 7 \).

Step4: Find \( y \) for \( x = 10 \)

Substitute \( x = 10 \) into \( y = 3x \):
\( y = 3(10) = 30 \).

Problem 9 (Equation of a Line in Slope-Intercept Form)

Step1: Recall slope-intercept form

Slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

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

The line passes through \( (0, 2) \). At \( x = 0 \), \( y = b \), so \( b = 2 \).

Step3: Calculate the slope (\( m \))

Using points \( (0, 2) \) and \( (-3, 14) \):
\( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{14 - 2}{-3 - 0} = \frac{12}{-3} = -4 \).

Step4: Write the equation

Substitute \( m = -4 \) and \( b = 2 \) into \( y = mx + b \):
\( y = -4x + 2 \).

Final Answers:

Problem 8 Table:

Number of Snacks, \( x \)	Cost (\$), \( y \)
4	12
7	21
10	30

Problem 9:

The equation of the line is \( \boldsymbol{y = -4x + 2} \).

Answer:

Step1: Recall slope-intercept form

Slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

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

The line passes through \( (0, 2) \). At \( x = 0 \), \( y = b \), so \( b = 2 \).

Step3: Calculate the slope (\( m \))

Using points \( (0, 2) \) and \( (-3, 14) \):
\( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{14 - 2}{-3 - 0} = \frac{12}{-3} = -4 \).

Step4: Write the equation

Substitute \( m = -4 \) and \( b = 2 \) into \( y = mx + b \):
\( y = -4x + 2 \).

Final Answers:

Problem 8 Table:

Number of Snacks, \( x \)	Cost (\$), \( y \)
4	12
7	21
10	30

Problem 9:

The equation of the line is \( \boldsymbol{y = -4x + 2} \).