Problem 24 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Passing through \((8,-2)\) and \((-2,3)\)

Step-by-Step Solution

Answer

The slope-intercept form is \(y = -\frac{1}{2}x + 2\).

Step 1: Identify the Given Points

The given points through which the line passes are \((8, -2)\) and \((-2, 3)\). Let's denote these points as \( (x_1, y_1) = (8, -2) \) and \( (x_2, y_2) = (-2, 3) \).

Step 2: Calculate the Slope

The formula to calculate the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substituting the given point values, we have: \[m = \frac{3 - (-2)}{-2 - 8} = \frac{5}{-10} = -\frac{1}{2}\]So, the slope \(m\) of the line is \(-\frac{1}{2}\).

Step 3: Use the Point-Slope Form

Now, use the point-slope form of the line to find the equation. The point-slope form is given by:\[y - y_1 = m(x - x_1)\]Using point \((8, -2)\) and slope \(-\frac{1}{2}\), substitute into the formula:\[y - (-2) = -\frac{1}{2}(x - 8)\]Simplify this equation to get the slope-intercept form.

Step 4: Simplify to Slope-Intercept Form

Begin by simplifying the equation from Step 3:\[y + 2 = -\frac{1}{2}(x - 8)\]Distribute the slope:\[y + 2 = -\frac{1}{2}x + 4\]Subtract 2 from both sides to solve for \(y\):\[y = -\frac{1}{2}x + 2\]Thus, the slope-intercept form of the line is \(y = -\frac{1}{2}x + 2\).

Key Concepts

Slope Calculation

When we talk about slope calculation in mathematics, we're essentially discussing how steep a line is on a graph. The slope can be thought of as a measure of how much the y-value of a line increases or decreases as the x-value increases.
To find the slope of a line passing through two points,

• use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
• where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

In our example, the points given are \((8, -2)\) and \((-2, 3)\).
By substituting these into the formula, you find that the slope \(m\) is \(-\frac{1}{2}\). This means that for each unit increase in the x direction, the y value decreases by half a unit.
This concept is crucial as it allows us to understand the trend and direction of the line on a graph.

Point-Slope Form

Once we know the slope of a line, we can use the point-slope form to write the equation of the line. The point-slope form is written as follows:

• \[ y - y_1 = m(x - x_1) \]
• where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

This form is particularly helpful when you know the slope and one point that the line passes through.
In our example, we used the point \((8, -2)\) and the slope \(-\frac{1}{2}\). By substituting these values into the point-slope form, we set up the equation \[ y + 2 = -\frac{1}{2}(x - 8) \].
This form is often used as a stepping stone to transform equations into the slope-intercept form, which is easier to use for graphing.

Linear Equations

Linear equations are equations that result in a straight line when plotted on a graph. These equations can be expressed in several forms, such as the point-slope form or the slope-intercept form.
A commonly used form is the slope-intercept form, which looks like this:

• \[ y = mx + b \]
• where \(m\) is the slope and \(b\) is the y-intercept (the point where the line crosses the y-axis).

In the solution, we converted from the point-slope form to the slope-intercept form by simplifying the equation.
Starting with \[ y + 2 = -\frac{1}{2}x + 4 \], subtracting 2 from both sides gives us \[ y = -\frac{1}{2}x + 2 \].
This final equation tells us that the line crosses the y-axis at \(y = 2\) and has a slope of \(-\frac{1}{2}\). This makes it straightforward to graph and understand the behavior of the line.