Problem 16 - Algebra A Combined Function

Question

Write an equation of the line with each given slope, \(m\), and \(y\) -intercept, \((0, b) .\) $$ m=2, b=\frac{3}{4} $$

Step-by-Step Solution

Answer

The equation of the line is \( y = 2x + \frac{3}{4} \).

Step 1: Understand the Equation of a Line

The general form of a linear equation in slope-intercept form is \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

Step 2: Substitute Given Values

You are given that the slope \( m = 2 \) and the y-intercept \( b = \frac{3}{4} \). Substitute these values into the general equation.

Step 3: Formulate the Specific Equation

Replacing \( m \) with 2 and \( b \) with \( \frac{3}{4} \), the equation becomes: \[y = 2x + \frac{3}{4}\]

Step 4: Validate the Equation

Ensure that the given values align with the formulated equation. The equation \( y = 2x + \frac{3}{4} \) correctly incorporates the slope and y-intercept provided in the problem.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a popular way to express straight lines. It is written as \( y = mx + b \). This form shows exactly where the line crosses the y-axis and how steep the line is.

Being in this format makes it straightforward to graph a line or understand its characteristics at a glance. For example, if you know both the slope \( m \) and the y-intercept \( b \), you can write the equation of the line immediately.

To write an equation for a line in this form, just plug the values you have directly into the standard equation as we did: \( y = mx + b \). This gives an efficient method to describe linear relationships in algebra.

Slope

Slope is a measure of how steep a line is. In the slope-intercept form \( y = mx + b \), the slope is represented by \( m \).

The slope tells you how much \( y \) changes for a given change in \( x \). Specifically, it's the change in the vertical direction divided by the change in the horizontal direction.

• A positive slope means the line goes upwards as you move from left to right.
• A negative slope means the line goes downwards as you move from left to right.
• A zero slope means the line is flat, indicating no vertical change.

For example, in the equation \( y = 2x + \frac{3}{4} \), the slope \( m \) is 2. This means for every 1 unit increase in \( x \), \( y \) increases by 2 units.

Y-Intercept

The y-intercept is where the line intersects the y-axis on a graph. In the equation \( y = mx + b \), the y-intercept is \( b \). This point occurs when \( x = 0 \).

The y-intercept provides a starting point for plotting the line. It shows where the line crosses the y-axis. For the given equation \( y = 2x + \frac{3}{4} \), the y-intercept is \( \frac{3}{4} \).

This means that when \( x \) is 0, \( y \) will equal \( \frac{3}{4} \). Knowing this, alongside the slope, allows for quick graph plotting without needing a table of values.