he equation of a line is given by y = mx + b . Solve for x in terms of y .(1 point)

Question

Responses

x=ym+b
x is equal to y over m plus b

x=y−bm
x is equal to the fraction with numerator y minus b and denominator m

x=y+bm
x is equal to the fraction with numerator y plus b and denominator m

x=ym−b

Answer 1

To solve for x in terms of y in the equation of a line \( y = mx + b \):

1. Start with the equation: \[ y = mx + b \]

2. Subtract \( b \) from both sides: \[ y - b = mx \]

3. Divide both sides by \( m \): \[ x = \frac{y - b}{m} \]

So the correct response is: x = y − b over m (x is equal to the fraction with numerator y minus b and denominator m).

Answer 2

Which of the following tables represent a linear function with the same slope as y = 2 - 3x ?

Select 2 that apply.

(2 points)
x 2 4 6 8 10
y 10 8 6 4 2

x 2 4 6 8 10
y 25 19 13 7 1

x 0 1 2 3 4
y 2 5 8 11 14

x 0 1 2 3 4
y 2 4 6 8 10

x 0 1 2 3 4
y 10 7 4 1 -2

x 2 4 6 8 10
y 4 10 16 22 28

Answer 3

The equation \( y = 2 - 3x \) can be rewritten in the slope-intercept form \( y = mx + b \), where the slope \( m \) is -3. Thus, we are looking for tables that represent a linear function with the same slope of -3.

To determine the slope of each table, we can calculate the slope between any two points \((x_1, y_1)\) and \((x_2, y_2)\) using the formula:

\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]

Let's analyze the provided tables:

1. Table 1: \[ \begin{array}{c|ccccc} x & 2 & 4 & 6 & 8 & 10 \ \hline y & 10 & 8 & 6 & 4 & 2 \ \end{array} \] Slope between (2, 10) and (4, 8): \[ \text{slope} = \frac{8 - 10}{4 - 2} = \frac{-2}{2} = -1 \]

2. Table 2: \[ \begin{array}{c|ccccc} x & 2 & 4 & 6 & 8 & 10 \ \hline y & 25 & 19 & 13 & 7 & 1 \ \end{array} \] Slope between (2, 25) and (4, 19): \[ \text{slope} = \frac{19 - 25}{4 - 2} = \frac{-6}{2} = -3 \]

3. Table 3: \[ \begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \ \hline y & 2 & 5 & 8 & 11 & 14 \ \end{array} \] Slope between (0, 2) and (1, 5): \[ \text{slope} = \frac{5 - 2}{1 - 0} = \frac{3}{1} = 3 \]

4. Table 4: \[ \begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \ \hline y & 2 & 4 & 6 & 8 & 10 \ \end{array} \] Slope between (0, 2) and (1, 4): \[ \text{slope} = \frac{4 - 2}{1 - 0} = \frac{2}{1} = 2 \]

5. Table 5: \[ \begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \ \hline y & 10 & 7 & 4 & 1 & -2 \ \end{array} \] Slope between (0, 10) and (1, 7): \[ \text{slope} = \frac{7 - 10}{1 - 0} = \frac{-3}{1} = -3 \]

6. Table 6: \[ \begin{array}{c|ccccc} x & 2 & 4 & 6 & 8 & 10 \ \hline y & 4 & 10 & 16 & 22 & 28 \ \end{array} \] Slope between (2, 4) and (4, 10): \[ \text{slope} = \frac{10 - 4}{4 - 2} = \frac{6}{2} = 3 \]

Conclusion:

The tables that represent a linear function with the same slope as \( y = 2 - 3x\) (slope -3) are:

• Table 2
• Table 5

Answer 4

The equation of Line 1: 4x - 2y = 16

The equation of Line 2: y = -4/5x + 2

The equation of Line 3: y - 4 = 3 (x + 5)

Which statement is true?

(1 point)
Responses

Line 3 has a decreasing graph
Line 3 has a decreasing graph

Line 2 has the biggest y-intercept
Line 2 has the biggest y-intercept

Line 2 has an increasing graph
Line 2 has an increasing graph

Line 1 has the greatest slope.
Line 1 has the greatest slope.

Line 3 has the greatest slope
Line 3 has the greatest slope

Line 1 has the biggest y-intercept

Answer 5

Let's analyze each line's equation and characteristics to determine which statement is true.

Line 1: \(4x - 2y = 16\)

First, we will convert it into slope-intercept form \(y = mx + b\):

1. Rearrange: \[ -2y = -4x + 16 \]
2. Divide by \(-2\): \[ y = 2x - 8 \]

• Slope: 2 (increasing)
• Y-intercept: -8

Line 2: \(y = -\frac{4}{5}x + 2\)

This is already in slope-intercept form:

• Slope: \(-\frac{4}{5}\) (decreasing)
• Y-intercept: 2

Line 3: \(y - 4 = 3(x + 5)\)

Convert to slope-intercept form:

1. Distribute the 3: \[ y - 4 = 3x + 15 \]
2. Add 4: \[ y = 3x + 19 \]

• Slope: 3 (increasing)
• Y-intercept: 19

Summary of attributes:

• Line 1:
  • Slope: 2
  • Y-intercept: -8
• Line 2:
  • Slope: -\(\frac{4}{5}\)
  • Y-intercept: 2
• Line 3:
  • Slope: 3
  • Y-intercept: 19

Evaluating the statements:

1. Line 3 has a decreasing graph: False (Line 3 is increasing).
2. Line 2 has the biggest y-intercept: False (Line 3 has the biggest y-intercept).
3. Line 2 has an increasing graph: False (Line 2 is decreasing).
4. Line 1 has the greatest slope: False (Line 3 has a greater slope).
5. Line 3 has the greatest slope: True (slope = 3 is the greatest).
6. Line 1 has the biggest y-intercept: False (Line 3 has the biggest y-intercept).

Conclusion:

The true statement is: Line 3 has the greatest slope.

Answer 6

Ryan is tracking how much money he has each week.

He started with $60.

He spent $5 per week for the first 6 weeks.

He spent $10 per week after week 6 until the money's all spent.

(4 points)
A. The graph of the scenario would be (Response area) because the rate of change changes.

B. The y-intercept for this scenario would be (Response area) meaning (Response area).

C. The rate of change is (Response area) because as the weeks increase, the money decreases.

responses
Ryan started with $60 in his account
positive
(0, 60)
(10, 6)
nonlinear
Ryan decreasing his $ each week
(6, 5)
Ryan took 10 weeks to spend all of his $
linear
(10,0)
negative
(0, 5)

Answer 7

Let's analyze the scenario carefully to fill in the responses for parts A, B, and C:

A. The graph of the scenario would be nonlinear because the rate of change changes.

• This is true because for the first 6 weeks, Ryan spends $5 per week (a constant rate), and after week 6, he spends $10 per week (a different constant rate). This change in spending rate means that the graph will not represent a single linear function; it will have segments of linear behavior but will not be globally linear throughout.

B. The y-intercept for this scenario would be (0, 60) meaning Ryan started with $60 in his account.

• The y-intercept is indeed where \(x = 0\) (the start of the tracking), and at that point, Ryan has $60.

C. The rate of change is negative because as the weeks increase, the money decreases.

• This is correct because as time progresses (weeks), the total money Ryan has is decreasing due to his spending, and thus the rate of change is negative.

Summary of Responses:

A. The graph of the scenario would be nonlinear because the rate of change changes.

B. The y-intercept for this scenario would be (0, 60) meaning Ryan started with $60 in his account.

C. The rate of change is negative because as the weeks increase, the money decreases.

Answer 8

The equation 7x + 5y = 100 is in standard form. This form tells us the x and y intercepts easily.

What are the x and y intercepts?

(1 point)
Responses

The line crosses the x-axis at (20, 0) and crosses the y-axis at (0, 100/7).
The line crosses the x-axis at (20, 0) and crosses the y-axis at (0, 100/7).

The line crosses the x-axis at (7, 0) and crosses the y-axis at (0, 5).
The line crosses the x-axis at (7, 0) and crosses the y-axis at (0, 5).

The line crosses the x-axis at (100/7, 0) and crosses the y-axis at (0, 20).
The line crosses the x-axis at (100/7, 0) and crosses the y-axis at (0, 20).

The line crosses the x-axis at (7/100, 0) and crosses the y-axis at (0, 1/20).
The line crosses the x-axis at (7/100, 0) and crosses the y-axis at (0, 1/20).

Answer 9

To find the x and y intercepts of the equation \(7x + 5y = 100\):

Finding the x-intercept:

To determine the x-intercept, we set \(y\) to 0 and solve for \(x\):

\[ 7x + 5(0) = 100 \] \[ 7x = 100 \] \[ x = \frac{100}{7} \]

So, the x-intercept is at the point \(\left(\frac{100}{7}, 0\right)\).

Finding the y-intercept:

To determine the y-intercept, we set \(x\) to 0 and solve for \(y\):

\[ 7(0) + 5y = 100 \] \[ 5y = 100 \] \[ y = \frac{100}{5} = 20 \]

So, the y-intercept is at the point \((0, 20)\).

Summary of intercepts:

• x-intercept: \(\left(\frac{100}{7}, 0\right)\)
• y-intercept: \((0, 20)\)

Correct Response:

Therefore, the correct statement is: The line crosses the x-axis at \((\frac{100}{7}, 0)\) and crosses the y-axis at \((0, 20)\).

Answer 10

Michele has $25. She plans to save $10 per week until she has a total of $75.

Which equation matches this scenario?

(1 point)
Responses

10x + y = 25
10x + y = 25

y - 55 = 10(x - 3)
y - 55 = 10(x - 3)

y + 4 = 10 (x - 65)
y + 4 = 10 (x - 65)

y = 10x - 25

Answer 11

To find the equation that matches Michele's savings scenario, let's first define the variables:

• Let \( y \) be the total amount of money Michele has after saving.
• Let \( x \) be the number of weeks she saves.

Michele starts with $25 and plans to save $10 each week. The goal is to reach a total of $75.

Formulating the Equation

1. Michele starts with $25, so we can express the total amount after \( x \) weeks as: \[ y = 25 + 10x \]

2. We want to find out when \( y \) reaches $75: \[ 75 = 25 + 10x \] Solving for \( x \): \[ 75 - 25 = 10x \] \[ 50 = 10x \] \[ x = 5 \] This means she will have $75 after 5 weeks.

Now let's check which of the given options can be rearranged into the form \( y = 25 + 10x \).

1. Option 1: \( 10x + y = 25 \)

   • Rearranging gives: \[ y = 25 - 10x \] (Does NOT match.)

2. Option 2: \( y - 55 = 10(x - 3) \)

   • Expanding gives: \[ y - 55 = 10x - 30 \] \[ y = 10x + 25 \] (Does NOT match.)

3. Option 3: \( y + 4 = 10(x - 65) \)

   • Expanding gives: \[ y + 4 = 10x - 650 \] \[ y = 10x - 654 \] (Does NOT match.)

4. Option 4: \( y = 10x - 25 \)

   • This is already in slope-intercept form, but if we wanted to move it around: \[ y = 10x - 25 \] (Does NOT match.)

Conclusion

None of the provided equations perfectly match Michele's scenario expressed as \( y = 25 + 10x \).

The closest rewrite consistent with the information is: \[ y = 10x + 25 \] However, since that is not an option, none of the options provided are accurate descriptions of her savings scenario. If an option that looks like \( y = 10x + 25 \) was provided, that would have been the correct answer.

You may want to check the options again, as the values suggested do not correspond to the initial scenario described.