Problem 417 Find the equation of a line with... [FREE SOLUTION]

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Chapter 4: Problem 417

Find the equation of a line with given slope and containing the given point. Write the equation in slope-intercept form. \(m=-\frac{1}{3},\) point (-9,-8)

Short Answer

Expert verified

The equation of the line is \( y = -\frac{1}{3}x - 11 \).

Step by step solution

- Understand the slope-intercept form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \] Where m = slope b = y-intercept

- Substitute the given values

We are given: \( m = -\frac{1}{3} \) and the point (-9, -8). Substitute these values into the slope-intercept form to find 'b'.

- Plug in the point and slope to find the intercept

Substitute \( x = -9 \), \( y = -8 \) and \( m = -\frac{1}{3} \) into the equation: \[ -8 = -\frac{1}{3}(-9) + b \]

- Simplify the equation

Calculate: \[ -8 = 3 + b \] Solve for 'b': \[ b = -8 - 3 \] \[ b = -11 \]

- Write the final equation

Substitute \( m = -\frac{1}{3} \) and \( b = -11 \) back into the slope-intercept form: \[ y = -\frac{1}{3}x - 11 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form

The slope-intercept form is a way of writing the equation of a straight line. It's very popular because it's simple to use and understand. The formula looks like this: \[ y = mx + b \] Here, 'm' stands for the slope, and 'b' stands for the y-intercept. Using this formula, you can easily identify these two key components of a line.

finding the y-intercept

The y-intercept is the point where the line crosses the y-axis. To find this, you need to know both the slope and a point that the line passes through. In our exercise, we're given the slope \( m = -\frac{1}{3} \) and a point (-9, -8) . We substitute these values into the slope-intercept form to find the y-intercept, 'b'. This process helps us complete the equation of the line.

substituting values in equations

Substituting values is a method to find unknown variables by plugging known values into an equation. In the given exercise, we substitute the values of the slope \( m = -\frac{1}{3} \) and the coordinates of the given point (-9, -8) into the slope-intercept formula: \[ y = mx + b \] This allows us to solve for 'b', which is the missing piece in our equation.

slope of a line

The slope of a line measures how steep the line is. It is calculated as the change in y (vertical change) over the change in x (horizontal change). Mathematically, it's expressed as: \[ m = \frac{rise}{run} \] In our exercise, the slope \( m = -\frac{1}{3} \) indicates that for every 3 units the line moves horizontally, it falls by 1 unit. Knowing the slope helps in finding the equation of the line and also understanding the line's steepness.

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91影视

Find the equation of a line with given slope and containing the given point. Write the equation in slope-intercept form. \(m=-\frac{1}{3},\) point (-9,-8)

Short Answer

Step by step solution

- Understand the slope-intercept form

- Substitute the given values

- Plug in the point and slope to find the intercept

- Simplify the equation

- Write the final equation

Key Concepts

slope-intercept form

finding the y-intercept

substituting values in equations

slope of a line

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