Problem 24 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ m=8,(4,0) $$

Step-by-Step Solution

Answer

Answer: The equation of the line in slope-intercept form is y = 8x - 32.

Step 1: Identify the given slope and point

We are given the slope of the line (m) as 8 and a point on the line as (4, 0).

Step 2: Apply the point-slope form of a linear equation

To find the y-intercept (b), we can use the point-slope form of a linear equation, which is given by: $$ y - y_{1} = m(x - x_{1}) $$ where (x1, y1) is a point on the line and m is the slope. Substitute the given point (4,0) and the slope (m = 8) into the point-slope equation: $$ y - 0 = 8(x - 4) $$

Step 3: Solve for y

Now, we solve for y in the equation: $$ y = 8(x - 4) $$ Distribute the slope (8) through the parenthesis: $$ y = 8x - 32 $$

Step 4: Write the final equation in slope-intercept form

We have found the equation of the line in slope-intercept form, which is: $$ y = 8x - 32 $$

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a useful tool to express a straight line. It's one of the most common ways to write the equation of a line because of its ability to directly represent the slope and the y-intercept. This form is expressed as \( y = mx + b \), where:

• \( m \) is the slope of the line, representing how steep the line is.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

Slope-intercept form is particularly handy because it allows you to quickly identify the slope and y-intercept without needing to manipulate additional points or equations.
When writing an equation in this form, you simply plug in the given slope and y-intercept. If you know a point on the line, this form helps you find the exact line equation with ease. Once you practice converting equations and data into this format, analyzing linear relationships becomes straightforward.

Point-Slope Form

The point-slope form is another way of expressing the equation of a line, especially useful when you know one point on the line and the slope. The equation in point-slope form is written as \( y - y_1 = m(x - x_1) \), where:

• \( (x_1, y_1) \) represents a specific point on the line.
• \( m \) is the slope of the line.

This form is particularly helpful for constructing an equation quickly when data is presented as a point and a slope.
In cases where you need to convert from point-slope form to slope-intercept form, you solve for \( y \). This essentially gives you the slope-intercept equation by isolating \( y \) on one side of the equation. Once you grasp the point-slope concept, it's very easy to switch between different forms of a linear equation.

Solving Equations

Solving equations is a foundational skill in algebra that involves finding the value of a variable that makes an equation true. When dealing with linear equations, solving often means arranging the equation so that you isolate the variable of interest, usually \( y \) or \( x \).To solve an equation, it's important to:

• Use operations such as addition, subtraction, multiplication, and division to simplify both sides.
• Apply distributive properties if necessary, as seen with the multiplication of terms in parentheses.
• Rearrange terms to isolate the variable.

For instance, if you have an equation like \( y = 8(x - 4) \), solving involves distributing to get \( y = 8x - 32 \).
Mastering the skill of solving equations enables you to handle a variety of algebraic problems accurately and efficiently.