Problem 62 - Algebra for College Students

Question

Find an equation of the line that goes through the given point and has the given slope. Give the answer in slope-intercept form. See Example 5 (-5, 150) with slope -30

Step-by-Step Solution

Answer

The equation of the line is \( y = -30x \).

Step 1 - Recall the slope-intercept form

The slope-intercept form of a line's equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2 - Substitute the known values

Substitute the given slope (\( m = -30 \)) and the point (-5, 150) into the equation. Note that the equation becomes \( 150 = -30(-5) + b \).

Step 3 - Solve for the y-intercept

Calculate the value of \( b \) by solving the equation \( 150 = -30(-5) + b \). This simplifies to \( 150 = 150 + b \), leading to \( b = 0 \).

Step 4 - Write the final equation

Now that both \( m \) and \( b \) are known, the equation of the line is \( y = -30x + 0 \), which simplifies to \( y = -30x \).

Key Concepts

slope-intercept form

The slope-intercept form is a way to write the equation of a line. In this form, the equation looks like this: \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) is the y-intercept. The slope tells us how steep the line is, while the y-intercept is where the line crosses the y-axis. The slope-intercept form is very useful because it allows us to quickly graph the line and understand its behavior.
For example, if \( m \) is positive, the line rises as it moves from left to right. If \( m \) is negative, it falls as it moves from left to right.

finding the y-intercept

To find the y-intercept in the equation of a line, we need to substitute the known values into the slope-intercept form. First, we plug in the coordinates of a point on the line, as well as the slope. Using the example from the exercise, we know the point is (-5, 150) and the slope is -30. Thus, we substitute these values into the equation:

\( 150 = -30(-5) + b \)
Now, solve for \( b \) by simplifying the equation:

\( 150 = 150 + b \)
Subtracting 150 from both sides gives us:

\( 0 = b \)
Therefore, the y-intercept \( b \) is 0.

solving equations

Solving equations involves finding the unknown value that makes the equation true. In this case, we solved for the y-intercept \( b \). We started with

\( 150 = -30(-5) + b \)
First, we simplified the multiplication, which resulted in:
\( 150 = 150 + b \)
To isolate \( b \), we needed to perform the inverse operation. Since 150 was added to \( b \), we subtracted 150 from both sides:
\( 150 - 150 = 150 + b - 150 \)
This simplified to:
\( 0 = b \)
This approach can be used to solve any linear equation by isolating the variable through inverse operations like addition, subtraction, multiplication, or division.
Understanding these steps is crucial when dealing with more complex equations as well.