Problem 108 - College Algebra

Question

Find the equation of the line that passes through the following points: \((2 a, b)\) and \((a, b+1)\).

Step-by-Step Solution

Answer

The equation is \(y = -\frac{1}{a}x + (b + 2)\).

Step 1: Identify the Points

We have two points given: - Point 1 is \((x_1, y_1) = (2a, b)\) - Point 2 is \((x_2, y_2) = (a, b+1)\).

Step 2: Calculate the Slope

The formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the given points into the formula:\[m = \frac{(b+1) - b}{a - 2a} = \frac{1}{-a} = -\frac{1}{a}\].

Step 3: Use the Point-Slope Form

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

Step 4: Simplify the Equation

Distribute and simplify the equation from the previous step:\[y - b = -\frac{1}{a}x + 2\]\[y = -\frac{1}{a}x + 2 + b\].

Step 5: Write the Final Equation

The equation of the line in slope-intercept form is:\[y = -\frac{1}{a}x + (b + 2)\].

Key Concepts

slope calculation

In algebra, understanding how to calculate the slope between two points is essential. The slope of a line measures its steepness and direction. It is calculated using two points, often represented as \((x_1, y_1)\) and \((x_2, y_2)\).
The formula for the slope \(m\) is:

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

Here, you subtract the \(y\)-coordinates to find the change in \(y\), and \(x\)-coordinates to find the change in \(x\).
Substituting the given points \((2a, b)\) and \((a, b+1)\) into the formula:

• \( m = \frac{(b+1) - b}{a - 2a} = \frac{1}{-a} = -\frac{1}{a} \)

This shows a negative slope, indicating the line decreases as it moves from left to right. Calculating the slope is a crucial first step towards finding the equation of a line.

point-slope form

Once the slope is known, the point-slope form is a powerful tool to express the equation of a line. It is especially helpful when you have one point and the slope.
The point-slope form is represented as:

• \( y - y_1 = m(x - x_1) \)

This equation uses one known point \((x_1, y_1)\) and the calculated slope \(m\) to develop a linear equation.
Using the slope \(-\frac{1}{a}\) and point \((2a, b)\), the equation becomes:

• \( y - b = -\frac{1}{a}(x - 2a) \)

This formula provides a straightforward way to begin writing the equation of a line.
It will eventually lead us to a more commonly used form, the slope-intercept form, through further simplification.

slope-intercept form

Simplifying a point-slope equation often leads to the slope-intercept form. This is the most recognized equation of a line, expressed as \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
Simplifying the equation from the point-slope form \(y - b = -\frac{1}{a}(x - 2a)\), you distribute and rearrange terms:

• \( y - b = -\frac{1}{a}x + 2 \)
• \( y = -\frac{1}{a}x + 2 + b \)
• \( y = -\frac{1}{a}x + (b + 2) \)

This rearrangement demonstrates how the slope \(m = -\frac{1}{a}\) and the y-intercept \((b+2)\) are integrated into the equation.
Recognizing different forms of linear equations is crucial for various algebraic applications.
The slope-intercept form is particularly useful for graphing and analyzing the behavior of lines.

algebra

Algebra serves as the foundational language of mathematics, enabling you to describe relationships between quantities.
At its core, it involves using symbols and letters to represent numbers and express mathematical ideas in general terms.
In working with equations of lines, algebra allows you to:

• Determine relationships between variables
• Solve for unknowns
• Transform equations to different forms

The process of calculating the slope, converting between point-slope and slope-intercept forms, and simplifying equations are all rooted in algebraic thinking.
These skills build upon the basic principles of algebra, showcasing its importance in widespread areas of mathematics and real-world problem solving.