Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, find the equation of the line using the point- slope formula. Write all the final equations using the slope-intercept form. (1,3)\(\quad\) and \(\quad(5,5)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = \frac{1}{2}x + \frac{5}{2}\).

Step 1: Identify Given Points

We are given two points on the line: \((1, 3)\) and \((5, 5)\). Each point is represented as \((x_1, y_1)\) and \((x_2, y_2)\) respectively.

Step 2: Calculate the Slope

The formula for the slope \(m\) of a line 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 points, we get: \[ m = \frac{5 - 3}{5 - 1} = \frac{2}{4} = \frac{1}{2} \] Thus, the slope of the line is \(\frac{1}{2}\).

Step 3: Use the Point-Slope Formula

The point-slope form of a linear equation is: \[ y - y_1 = m(x - x_1) \] Using the point \((1, 3)\) and slope \(m = \frac{1}{2}\), the equation becomes: \[ y - 3 = \frac{1}{2}(x - 1) \]

Step 4: Convert to Slope-Intercept Form

First, distribute the slope on the right side of the equation: \[ y - 3 = \frac{1}{2}x - \frac{1}{2} \] Next, add 3 to both sides to solve for \(y\): \[ y = \frac{1}{2}x - \frac{1}{2} + 3 \] Simplify the expression: \[ y = \frac{1}{2}x + \frac{5}{2} \] This is the slope-intercept form \(y = mx + b\), where the slope \(m\) is \(\frac{1}{2}\) and the y-intercept \(b\) is \(\frac{5}{2}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most popular ways to express a linear equation. It's perfect for quickly identifying the slope and y-intercept of a line. The general formula for this form is \( y = mx + b \), where:

• \( m \) is the slope, indicating how steep the line is.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

The great thing about this form is its simplicity and straightforwardness. If you have the slope and the y-intercept, you can immediately construct the equation of the line. For example, in our exercise, we found the slope to be \( \frac{1}{2} \) and the y-intercept to be \( \frac{5}{2} \), so the slope-intercept form is \( y = \frac{1}{2}x + \frac{5}{2} \). This format is useful for graphing lines quickly or understanding how different lines compare just by looking at equations.

Linear Equations

Linear equations form the backbone of algebra, representing relationships in which the graph of the solutions forms a straight line. These equations can often involve two variables and manifest in various forms, such as standard form, point-slope form, and slope-intercept form.
Linear equations follow the pattern of constant change - as one variable increases or decreases, the other changes proportionally. Think of them as the MVPs of algebra, constantly showing up across different math problems and real-life applications. They can predict trends, map courses, and provide insights into relationships between quantities.
When working with linear equations, it is crucial to understand the relationship between the variables involved and how alterations in their values affect the equation as a whole. Converting between different forms, as we did in our exercise, showcases the flexibility and versatility of linear equations.

Slope Calculation

Calculating the slope of a line is about understanding its incline or steepness. The slope tells us how much the y-value of a line changes for a certain amount of change in the x-value. It's crucial for determining how steep or flat a line is. The formula for calculating slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In our exercise, using the points \((1, 3)\) and \((5, 5)\), we substituted to get \( m = \frac{5 - 3}{5 - 1} = \frac{1}{2} \). Our calculated slope of \( \frac{1}{2} \) signifies that for every increase of 2 in x, y increases by 1.
Understanding slope is vital in characterizing lines and understanding their behavior. It tells us whether a line rises, falls, or remains constant as we move along the x-axis. This knowledge is not only fundamental in mathematics but also in fields such as physics, engineering, and economics, where interpreting trends is essential.