Write the equation of the line that passes through the points (7, –4) and (–1, 3), first in point-slope form, and then in slope-intercept form.

Question

The slope of the line is
.

When the point (7, –4) is used, the point-slope form of the line is
.

The slope-intercept form of the line is
.

Answer 1

To find the equation of the line passing through the points (7, –4) and (–1, 3), we first need to calculate the slope (m) using the formula:

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

Here, we can assign:

• Point 1: \((x_1, y_1) = (7, -4)\)
• Point 2: \((x_2, y_2) = (-1, 3)\)

Plugging in the coordinates:

\[ m = \frac{3 - (-4)}{-1 - 7} = \frac{3 + 4}{-1 - 7} = \frac{7}{-8} = -\frac{7}{8} \]

So, the slope of the line is \(-\frac{7}{8}\).

Point-Slope Form

The point-slope form of a line is given by the equation:

\[ y - y_1 = m(x - x_1) \]

Using the point (7, -4):

\[ y - (-4) = -\frac{7}{8}(x - 7) \]

This simplifies to:

\[ y + 4 = -\frac{7}{8}(x - 7) \]

So the point-slope form of the line is

\[ y + 4 = -\frac{7}{8}(x - 7) \]

Slope-Intercept Form

To convert to slope-intercept form (y = mx + b), we can distribute and isolate y. Starting from the point-slope form:

\[ y + 4 = -\frac{7}{8}(x - 7) \]

Distributing the slope:

\[ y + 4 = -\frac{7}{8}x + \frac{49}{8} \]

Now, subtract 4 from both sides. We convert 4 to a fraction with a denominator of 8:

\[ 4 = \frac{32}{8} \]

So we have:

\[ y = -\frac{7}{8}x + \frac{49}{8} - \frac{32}{8} \]

Combining the constant terms gives:

\[ y = -\frac{7}{8}x + \frac{17}{8} \]

Thus, the slope-intercept form of the line is

\[ y = -\frac{7}{8}x + \frac{17}{8} \]

Summary

• The slope of the line is \(-\frac{7}{8}\).
• The point-slope form of the line is \(y + 4 = -\frac{7}{8}(x - 7)\).
• The slope-intercept form of the line is \(y = -\frac{7}{8}x + \frac{17}{8}\).

Answer 2

Write the equation of the line that passes through the points (8, –1) and (2, –5) in standard form, given that the point-slope form is y + 1 = (x – 8).

___x +___
y=___

Answer 3

To find the equation of the line passing through the points \( (8, -1) \) and \( (2, -5) \) in standard form (Ax + By = C), let's first calculate the slope of the line using the formula:

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

Assigning the points:

• Point 1: \( (x_1, y_1) = (8, -1) \)
• Point 2: \( (x_2, y_2) = (2, -5) \)

Calculating the slope:

\[ m = \frac{-5 - (-1)}{2 - 8} = \frac{-5 + 1}{2 - 8} = \frac{-4}{-6} = \frac{2}{3} \]

Now we know the slope \( m = \frac{2}{3} \).

Point-Slope Form

We can start from the point-slope form, which is given by:

\[ y - y_1 = m(x - x_1) \]

Using point (8, -1):

\[ y - (-1) = \frac{2}{3}(x - 8) \]

This simplifies to:

\[ y + 1 = \frac{2}{3}(x - 8) \]

Converting to Standard Form

To convert this to standard form \( Ax + By = C \), we will first simplify the equation and then rearrange it.

1. Distributing the slope:

\[ y + 1 = \frac{2}{3}x - \frac{16}{3} \]

2. Subtracting 1 (which is \( \frac{3}{3} \) to have a common denominator):

\[ y = \frac{2}{3}x - \frac{16}{3} - \frac{3}{3} \]

\[ y = \frac{2}{3}x - \frac{19}{3} \]

3. To eliminate the fraction, multiply every term by 3:

\[ 3y = 2x - 19 \]

4. Rearranging to standard form:

\[ -2x + 3y = -19 \]

or by multiplying through by -1 to make the leading coefficient positive:

\[ 2x - 3y = 19 \]

Final Answer

In standard form, the equation of the line is:

\[ 2x - 3y = 19 \]

So, filling in the blanks:

x + y = ___

This translates to:

• \(2\) x + \(-3\) y = \(19\)

Answer 4

Write the equation of a line that passes through the points shown in the table.Which equations represent a line that passes through the points given in the table? Check all that apply.y – 2 = –6(x + 10)y – 2 = –(x + 10)y – 1 = –(x + 4)y = –6x – 58y = –x + y = –x + 5

x | y
-10| 2
-4| 1
8| -1
14| -2

Answer 5

To determine which equations represent a line that passes through the points given in the table, we'll first find the slope of the line connecting the points and identify the y-intercept, if possible. The points from the table are:

1. \( (-10, 2) \)
2. \( (-4, 1) \)
3. \( (8, -1) \)
4. \( (14, -2) \)

Step 1: Calculate the slope

Let's use two points, for instance, \( (-10, 2) \) and \( (-4, 1) \), to calculate the slope \( m \):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 2}{-4 - (-10)} = \frac{1 - 2}{-4 + 10} = \frac{-1}{6} = -\frac{1}{6} \]

So, the slope of the line is \( -\frac{1}{6} \).

Step 2: Use point-slope form to determine the linear equation

We can use point-slope form \( y - y_1 = m(x - x_1) \):

Using the point \( (-10, 2) \):

\[ y - 2 = -\frac{1}{6}(x + 10) \]

Step 3: Convert to slope-intercept form

Expanding this:

\[ y - 2 = -\frac{1}{6}x - \frac{10}{6} \]

Adding 2 (or \( \frac{12}{6} \)) to both sides:

\[ y = -\frac{1}{6}x - \frac{10}{6} + \frac{12}{6} \] \[ y = -\frac{1}{6}x + \frac{2}{6} \] \[ y = -\frac{1}{6}x + \frac{1}{3} \]

Step 4: Check the given equations

Now let's analyze the given equations. We’ll check if any of these match the slope we found, which is \( -\frac{1}{6} \).

1. \( y - 2 = -6(x + 10) \):

   • Rearranging gives \( y = -6x - 60 + 2 = -6x - 58\). (Does not match)

2. \( y - 2 = -(x + 10) \):

   • Rearranging gives \( y = -x - 10 + 2 = -x - 8\). (Does not match)

3. \( y - 1 = -(x + 4) \):

   • Rearranging gives \( y = -x - 4 + 1 = -x - 3\). (Does not match)

4. \( y = -6x - 58 \): (Does not match)

5. \( y = -x + 5 \): (Does not match)

Conclusion:

None of the provided equations went through the calculations we derived based on the points given.

Summary of results:

• None of the equations given match the line that goes through the points in the table.

If you need to test any more equations or need further clarification, feel free to ask!

Answer 6

what? there must be something right?

here sorry i had to rewrite some stuff Write the equation of a line that passes through the points shown in the table.Which equations represent a line that passes through the points given in the table? Check all that apply.y – 2 = –6(x + 10)y – 2 = –1/6 (x + 10)y – 1 =1/6 –(x + 4)y = –6x – 58 y = –1/6x +1/3 y = –1/6x + 5

Answer 7

Let's determine which equations represent a line that passes through the points you've provided in the table. The points are:

• \( (-10, 2) \)
• \( (-4, 1) \)
• \( (8, -1) \)
• \( (14, -2) \)

Step 1: Calculate the slope

To start, we will calculate the slope between two given points. Let's use the points \( (-10, 2) \) and \( (-4, 1) \):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 2}{-4 - (-10)} = \frac{-1}{-4 + 10} = \frac{-1}{6} = -\frac{1}{6} \]

The slope \( m = -\frac{1}{6} \).

Step 2: Point-Slope Form to Find the Equation

Using the point-slope form of a line equation \( y - y_1 = m(x - x_1) \) with the point \( (-10, 2) \):

\[ y - 2 = -\frac{1}{6}(x + 10) \]

Step 3: Check the Provided Equations

Now we need to check which of the given equations represents a line with this slope. Here are the equations you provided:

1. \( y - 2 = -6(x + 10) \)

   Rearranging gives:

   \[ y - 2 = -6x - 60 \implies y = -6x - 58 \]

   (Slope: -6) — Not a match.

2. \( y - 2 = -\frac{1}{6}(x + 10) \)

   Rearranging gives:

   \[ y - 2 = -\frac{1}{6}x - \frac{10}{6} \implies y = -\frac{1}{6}x - \frac{10}{6} + 2 \] \[ y = -\frac{1}{6}x - \frac{10}{6} + \frac{12}{6} = -\frac{1}{6}x + \frac{2}{6} = -\frac{1}{6}x + \frac{1}{3} \]

   (Slope: -\frac{1}{6}) — This is a match.

3. \( y - 1 = \frac{1}{6}( - (x + 4)) \)

   Rearranging gives:

   \[ y - 1 = -\frac{1}{6}(x + 4) \implies y - 1 = -\frac{1}{6}x - \frac{4}{6} \] \[ y = -\frac{1}{6}x + 1 - \frac{2}{3} \] \[ y = -\frac{1}{6}x + \frac{3}{3} - \frac{2}{3} = -\frac{1}{6}x + \frac{1}{3} \]

   (Slope: -\frac{1}{6}) — This is a match.

4. \( y = -6x - 58 \)

   (Slope: -6) — Not a match.

5. \( y = -\frac{1}{6}x + \frac{1}{3} \)

   (Slope: -\frac{1}{6}) — This is a match.

6. \( y = -\frac{1}{6}x + 5 \)

   (Slope: -\frac{1}{6}) — This is a match.

Summary of Results

The equations that represent a line that passes through the points given in the table are:

1. \( y - 2 = -\frac{1}{6}(x + 10) \)
2. \( y - 1 = -\frac{1}{6}(x + 4) \)
3. \( y = -\frac{1}{6}x + \frac{1}{3} \)
4. \( y = -\frac{1}{6}x + 5 \)

Thus, we have four valid equations from those provided that pass through the points given in the table.

Answer 8

**\( y = -\frac{1}{6}x + 5 \)** is not correct

After 2 hours, there are 1,400 mL of fluids remaining in a patient’s IV. The fluids drip at a rate of 300 mL per hour. Let x be the time passed, in hours, and y be the amount of fluid left in the IV, in mL. Write a linear function that models this scenario.



The slope of the line is
.

The y-intercept of the line is
.

The linear function is
.

Answer 9

To model the scenario of a patient’s IV fluids, we can define the variables and find the linear function based on the information given:

• Let \( x \) be the time passed in hours.
• Let \( y \) be the amount of fluid left in the IV in mL.

Step 1: Determine the Slope

In this scenario, the fluids drip at a rate of 300 mL per hour. This means that for every hour that passes, the amount of fluid decreases by 300 mL. Therefore, the slope \( m \) of the line is:

\[ m = -300 , \text{mL/hour} \]

Step 2: Determine the Y-Intercept

The y-intercept is the initial amount of fluid when \( x = 0 \). According to the problem, at time \( x = 0 \) hours, there are 1,400 mL of fluids remaining. Thus, the y-intercept \( b \) is:

\[ b = 1400 , \text{mL} \]

Step 3: Write the Linear Function

The linear function can be represented in the slope-intercept form, which is:

\[ y = mx + b \]

Substituting the values of the slope and y-intercept into this equation gives:

\[ y = -300x + 1400 \]

Summary

• The slope of the line is -300.
• The y-intercept of the line is 1400.
• The linear function is

\[ y = -300x + 1400 \]

This function models the amount of fluid left in the IV as time passes.

Answer 10

The slope of the line is
✔ –300
.

The y-intercept of the line is
✔ 2,000
.

The linear function is
✔ f(x) = –300x + 2,000
.

Pedro works as a newspaper deliverer. He receives a base pay plus an additional amount per newspaper delivered. Last week, Pedro delivered 5 newspapers and earned $37.50. This week, he delivered 20 newspapers and earned $75. Let x equal the number of newspapers Pedro delivers and y be the earnings he receives in dollars. Which linear function models the scenario?

g(x) = 0.4x + 35.5
g(x) = 0.4x + 67
g(x) = 2.5x + 25
g(x) = 2.5x + 50

Answer 11

To find the linear function that models Pedro's earnings based on the number of newspapers delivered, we need to establish a relationship between the number of newspapers delivered (\(x\)) and his earnings (\(y\)).

We have two pieces of information:

1. Last week, Pedro delivered 5 newspapers and earned $37.50.
2. This week, he delivered 20 newspapers and earned $75.

Step 1: Create Two Points

From the information given, we can represent the two situations as points in the form \((x, y)\):

1. Point 1: \((5, 37.50)\)
2. Point 2: \((20, 75)\)

Step 2: Calculate the Slope

We can calculate the slope (\(m\)) of the line using the formula:

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

Using the points:

• Point 1: \((x_1, y_1) = (5, 37.50)\)
• Point 2: \((x_2, y_2) = (20, 75)\)

Calculating the slope:

\[ m = \frac{75 - 37.50}{20 - 5} = \frac{37.50}{15} = 2.5 \]

So, the slope \(m\) is \(2.5\).

Step 3: Determine the Y-Intercept

Now we can use one of the points to find the y-intercept \(b\). We can use the slope-intercept form of the equation:

\[ y = mx + b \]

Substituting \(m = 2.5\) and using point \((5, 37.50)\):

\[ 37.50 = 2.5(5) + b \]

Calculating:

\[ 37.50 = 12.5 + b \]

Now, solving for \(b\):

\[ b = 37.50 - 12.5 = 25 \]

Step 4: Write the Linear Function

Now we can write the linear function that models Pedro's earnings:

\[ y = 2.5x + 25 \]

Or in function notation:

\[ g(x) = 2.5x + 25 \]

Check which equation matches

From your options:

1. g(x) = 0.4x + 35.5
2. g(x) = 0.4x + 67
3. g(x) = 2.5x + 25
4. g(x) = 2.5x + 50

The correct linear function that models the scenario is:

g(x) = 2.5x + 25.

Conclusion

Thus, Pedro's earnings can be modeled by the equation: \(g(x) = 2.5x + 25\).