Problem 3 - A Graphical Approach to College Algebra

Question

Write the slope-intercept form of the line that passes through the given point with slope \(m .\) Do not use a calculator. Through \((-5,4), m=1.5\)

Step-by-Step Solution

Answer

The equation of the line is \(y = 1.5x + 11.5\).

Step 1: Recall the Slope-Intercept Form

The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Our goal is to find the equation of the line in this form.

Step 2: Substitute the Given Slope

We are given that the slope \(m = 1.5\). Substitute this value into the slope-intercept form: \(y = 1.5x + b\).

Step 3: Use the Given Point to Find the Intercept

The line passes through the point \((-5, 4)\). Substitute \(x = -5\) and \(y = 4\) into the equation to find \(b\):\[4 = 1.5(-5) + b\]

Step 4: Solve for the Y-Intercept

Simplify and solve for \(b\):\[4 = -7.5 + b\]\[b = 4 + 7.5 = 11.5\]So, the y-intercept \(b\) is 11.5.

Step 5: Write the Final Equation

Now substitute \(m = 1.5\) and \(b = 11.5\) back into the slope-intercept form: \[y = 1.5x + 11.5\]This is the final equation of the line in slope-intercept form.

Key Concepts

Slope

The slope of a line is a measure of its steepness and direction. When you think about the slope, imagine a hill: the steeper the hill, the larger the slope. In the context of linear equations, the slope is crucial for defining the angle at which a line tilts.

In mathematical terms, slope is represented by the letter \(m\), and it is calculated as the "rise over run." This means it is the change in \(y\) (vertical direction) divided by the change in \(x\) (horizontal direction) between any two points on a line. A positive slope indicates that the line rises as it moves from left to right, while a negative slope indicates the line falls. A zero slope means the line is horizontal, and an undefined slope means the line is vertical.

In the problem you're working with, the slope \(m\) is 1.5, which means for every unit the line moves horizontally, it moves up 1.5 units vertically. This information helps us figure out how to draw the line on a graph.

Y-Intercept

The y-intercept is another key element in the linear equation. It is the point where the line crosses the y-axis. You can find this by looking at the equation of a line in slope-intercept form, \(y = mx + b\), where \(b\) represents the y-intercept.

Understanding the y-intercept is like knowing where a ball lands if you roll it from a slope onto a flat road. It's the starting point on the y-axis when \(x\) is zero. In any graph of a line, this is the coordinate \((0, b)\).

In the exercise provided, after substituting the given point and solving, we found the y-intercept \(b\) to be 11.5. This means when \(x = 0\), the line crosses the y-axis at the point \((0, 11.5)\). It helps in drawing the line because you know a specific point through which the line passes.

Equation of a Line

Forming the equation of a line in slope-intercept form ties together understanding of both slope and y-intercept. This form is written as \(y = mx + b\). Here, \(m\) is the slope, and \(b\) is the y-intercept. This format allows anyone to quickly identify how a line behaves on a graph just by looking at the equation.

To create the equation of a line that passes through a specific point, like the one in the exercise, you can begin by plugging the given slope value into the formula. Then, use the coordinates of the point (in this case, \((-5, 4)\)) to solve for \(b\). Finally, substitute both \(m\) and \(b\) back into the equation.

For the specific exercise, after inserting the slope \(m = 1.5\) and calculating the y-intercept \(b = 11.5\), the full equation is obtained as \(y = 1.5x + 11.5\). This equation can be used to plot the line or to find other points that lie on that line, showcasing how the concepts of slope and y-intercept coalesce to form a complete understanding of linear equations.