Problem 17 - A Graphical Approach to College Algebra

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. Through \((3,-8)\) and \((5,-3)\)

Step-by-Step Solution

Answer

The line's equation is \(y = \frac{5}{2}x - \frac{31}{2}\).

Step 1: Understand the Slope Formula

To find the slope-intercept form of a line, we need to first find the slope. The slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). In this problem, we substitute the points \((3, -8)\) and \((5, -3)\) into the formula.

Step 2: Calculate the Slope

Using the formula from Step 1 and the points \((3, -8)\) and \((5, -3)\), we calculate the slope: \(m = \frac{-3 - (-8)}{5 - 3} = \frac{-3 + 8}{2} = \frac{5}{2}\). Thus, the slope of the line is \(\frac{5}{2}\).

Step 3: Use the Point-Slope Formula

Now that we have the slope, we can use the point-slope form of the line equation: \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Substitute \((3, -8)\) and the slope \(\frac{5}{2}\) into this formula: \(y + 8 = \frac{5}{2}(x - 3)\).

Step 4: Convert to Slope-Intercept Form

We need to convert the equation from Step 3 into slope-intercept form \(y = mx + b\). Start by distributing \(\frac{5}{2}\): \(y + 8 = \frac{5}{2}x - \frac{15}{2}\). Subtract 8 from both sides to solve for \(y\): \(y = \frac{5}{2}x - \frac{15}{2} - 8\). Convert 8 to have a common denominator: \(8 = \frac{16}{2}\), so the equation becomes \(y = \frac{5}{2}x - \frac{31}{2}\).

Step 5: Finalize the Equation

The slope-intercept form of the line is achieved: \(y = \frac{5}{2}x - \frac{31}{2}\). This is the final equation of the line in slope-intercept form.

Key Concepts

Slope Formula

The Slope Formula is an essential concept when working with linear equations. It provides a way to determine the steepness of a line connecting two points on a Cartesian plane.
The slope, often denoted as \(m\), is calculated using 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.

This formula gives the rate of change between the two points, or simply, how much \(y\) changes with a change in \(x\). That means that for each unit that \(x\) increases, \(y\) increases by \(m\) times.

In the original exercise, using points \((3, -8)\) and \((5, -3)\), the slope is calculated by substituting these points into the formula, resulting in a slope of \(\frac{5}{2}\). This tells us that for every 2 units increase in \(x\), \(y\) increases by 5 units.

Point-Slope Form

Understanding the Point-Slope Form of an equation is crucial for writing linear equations given a point on the line and its slope.
The formula is:\[y - y_1 = m(x - x_1)\]where:

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

This form is particularly handy as it allows one to write the equation of a line quickly given a point and the slope.

In the exercise, the point \((3, -8)\) and the slope \(\frac{5}{2}\) are used to substitute in the formula, resulting in \(y + 8 = \frac{5}{2}(x - 3)\).

The Point-Slope Form is a great stepping stone to shifting the equation into other forms, such as the Slope-Intercept Form, which is more common for graphing purposes.

Linear Equations

Linear Equations are at the heart of algebra and can describe straight lines on a graph.
They are generally expressed in the slope-intercept form:\[y = mx + b\]where:

• \(m\) is the slope.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

This form makes it easy to graph a line by simply plotting the y-intercept and using the slope to find another point. The beauty of linear equations lies in their simplicity and utility in various mathematical and real-world contexts.

In the problem discussed, the equation \(y = \frac{5}{2}x - \frac{31}{2}\) is derived, clearly displaying these components: a slope of \(\frac{5}{2}\) and an intercept of \(-\frac{31}{2}\). Such expressions enable easy calculation and visualization, providing a cornerstone for higher-level math concepts.