Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(13-26,\) begin by solving the linear equation for \(y .\) This will put the equation in slope-intercept form. Then find the slope and the \(y\) -intercept of the line with this equation. $$4 x+3 y=4$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \( y = -\frac{4}{3}x + \frac{4}{3} \). The slope of the line is \( -\frac{4}{3} \) and the y-intercept is \( \frac{4}{3} \).

Step 1: Isolate y

First, isolate \( y \) in the equation. This is done in two steps. First, subtract \( 4x \) from both sides of the equation: \( 3y = -4x + 4 \). Then, divide all terms by 3 to solve for \( y \): \( y = -\frac{4}{3}x + \frac{4}{3} \).

Step 2: Identify the slope

In the equation \( y = -\frac{4}{3}x + \frac{4}{3} \), the coefficient of \( x \) is \( -\frac{4}{3} \). Therefore, the slope \( m \) of the line is \( -\frac{4}{3} \).

Step 3: Identify the y-intercept

In the equation \( y = -\frac{4}{3}x + \frac{4}{3} \), the constant term is \( \frac{4}{3} \). Therefore, the y-intercept \( b \) of the line is \( \frac{4}{3} \).

Key Concepts

Solving Linear Equations

Solving linear equations is a foundational skill in algebra that allows you to find the value of an unknown variable. A linear equation looks like a straight line when graphed and can be written in various forms, such as slope-intercept form, point-slope form, or standard form.

When you're given an equation like \(4x + 3y = 4\), the goal is to rearrange it so that you have \(y\) on one side, revealing its relationship to \(x\). To solve this, you can use the following steps:

• Move all terms containing \(x\) to the other side of the equation by either addition or subtraction.
• Divide every term by the coefficient of \(y\) to isolate \(y\) on its own.

Ultimately, you aim to get the equation in the form of \(y = mx + b\), which is the slope-intercept form and reveals both the slope and the y-intercept of the line.

Slope of a Line

The slope of a line is a measure of its steepness and direction. In mathematics, the slope is often denoted by \(m\) and it represents the rate at which the line rises or falls as it moves along the x-axis.

In the slope-intercept form \(y = mx + b\), \(m\) is the number that multiplies \(x\). To find the slope:

• Identify the coefficient of \(x\) in the equation.
• If the coefficient is a fraction, it can tell you how many units up or down the line goes for each unit it goes to the right or left.
• A positive slope means the line is going up as it moves to the right, and a negative slope means it goes down.

For the given equation \(y = -\frac{4}{3}x + \frac{4}{3}\), the slope is \(-\frac{4}{3}\), indicating the line falls 4 units for every 3 units it moves to the right.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In other words, it's the value of \(y\) when \(x = 0\). This point is significant because it is where the function starts or originates when graphed on a coordinate plane.

In the slope-intercept equation \(y = mx + b\), the y-intercept is represented by \(b\), the constant term.

• To find the y-intercept, look for the term that does not contain an \(x\).
• This value indicates where the line will cross the y-axis.
• If the equation is properly solved for \(y\), this term will be plainly visible without any further calculation.

In the provided equation, \(y = -\frac{4}{3}x + \frac{4}{3}\), \(\frac{4}{3}\) is the y-intercept, meaning the line will cross the y-axis at the point \((0, \frac{4}{3})\).