Problem 26 - Elementary Algebra

Question

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=4 \text { and } b=-3 $$

Step-by-Step Solution

Answer

The equation is \( y = 4x - 3 \).

Step 1: Understanding the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the \( y \)-intercept. Our task is to substitute the given values of \( m \) and \( b \) into this equation.

Step 2: Substitute the Given Values

We are given \( m = 4 \) and \( b = -3 \). Substitute these values into the slope-intercept form equation: \( y = mx + b \). This becomes \( y = 4x - 3 \).

Step 3: Write the Final Equation

After substitution, the equation of the line in slope-intercept form is \( y = 4x - 3 \). Therefore, this is the equation of the line with the given slope and \( y \)-intercept.

Key Concepts

Equation of a Line

An equation of a line describes all the points along that line on a graph. It tells us the relationship between the x and y coordinates of any point on that line.
When dealing with the equation of a line, one of the most common forms used is the slope-intercept form. This form is expressed as \( y = mx + b \).

• \( y \): the variable representing the dependent value or the vertical axis
• \( x \): the variable representing the independent value or the horizontal axis
• \( m \): the slope of the line, which describes how steep the line is
• \( b \): the y-intercept, the point where the line crosses the y-axis

The slope-intercept form is particularly useful because it easily informs us of both the slope and the y-intercept of the line. This allows us to quickly plot the line on a graph by starting at the y-intercept and using the slope to find other points on the line.

Slope

The slope of a line, represented by the letter \( m \), is a crucial component of the equation of a line. It measures the steepness and direction of the line.
The formula for the slope is \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in the y-values and \( \Delta x \) is the change in the x-values between two points on the line.

• A positive slope means the line is rising as it moves from left to right across the graph.
• A negative slope indicates the line is falling as it moves from left to right.
• A zero slope means the line is horizontal and does not rise or fall.

In our original exercise, the given slope \( m \) was 4, meaning for every step to the right on the x-axis, the line rises 4 steps on the y-axis. This gives us insight into how quickly y increases as x increases.

y-intercept

The y-intercept of a line, noted with the letter \( b \), is where the line crosses the y-axis. It is a point on the graph where the value of \( x \) is zero.
This intercept is very important as it provides a starting point for drawing the line on a graph.

• In the slope-intercept form \( y = mx + b \), the term \( b \) tells us exactly where the line intersects the y-axis.
• If \( b \) is positive, the line crosses the y-axis above the origin (0,0).
• If \( b \) is negative, the line crosses below the origin.

In the exercise, the y-intercept was given as \( b = -3 \), meaning the line crosses the y-axis at the point (0, -3). This provides a precise starting point to draw our line with the slope provided.