Problem 23 - 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. $$3 x+2 y=3$$

Step-by-Step Solution

Answer

The slope is \(-\frac{3}{2}\) and the y-intercept is \frac{3}{2}.

Step 1: Convert to slope-intercept form

Begin by subtracting \(3x\) from both sides of the equation \(3x + 2y = 3\) to isolate \(2y\). This gives us \(2y = -3x + 3\). Then, divide the entire equation by \(2\) to solve for \(y\). This gives us \(y = -\frac{3x}{2} + \frac{3}{2}\). The equation is now in the form \(y = mx + b\).

Step 2: Identify the Slope

The slope \(m\) is the coefficient of \(x\) in the equation \(y = mx + b\). Looking at our equation \(y = -\frac{3x}{2} + \frac{3}{2}\), the slope \(m\) is \(-\frac{3}{2}\).

Step 3: Identify the y-intercept

The \(y\)-intercept \(b\) is the constant in the equation \(y = mx + b\). Looking at our equation \(y = -\frac{3x}{2} + \frac{3}{2}\), the \(y\)-intercept \(b\) is \frac{3}{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 is written as \( y = mx + b \), where:

• \( y \) represents the dependent variable
• \( m \) is the slope of the line
• \( x \) is the independent variable
• \( b \) is the y-intercept

The beauty of using the slope-intercept form is that it provides immediate insight into the properties of the line. The slope \( m \) tells us how steep the line is, while the y-intercept \( b \) indicates where the line crosses the y-axis.

To convert a linear equation into this form, you need to solve the equation for \( y \). This involves isolating \( y \) on one side of the equation. By doing this, you can easily identify the slope and intercept from the resulting equation.

Solving for y

To put a linear equation into slope-intercept form, we must first solve for \( y \). Starting with an equation like \( 3x + 2y = 3 \), the goal is to isolate \( y \):

• Start by moving terms involving \( x \) to the other side of the equation: \( 2y = -3x + 3 \)
• Next, divide each term by the coefficient in front of \( y \) to solve for \( y \). In our example, divide by 2: \( y = -\frac{3}{2}x + \frac{3}{2} \)

Once \( y \) is isolated, the equation is in slope-intercept form. Now, it is straightforward to read the slope and intercept directly from the equation.

Remember, the process involves both operations of addition or subtraction to remove terms from one side, and division to adjust the coefficient of \( y \) to be 1.

Identifying Slope and Intercept

Once the linear equation is in the slope-intercept form \( y = mx + b \), finding the slope and y-intercept is simple:

• The slope \( m \) is the coefficient of \( x \). In \( y = -\frac{3}{2}x + \frac{3}{2} \), the slope \( m \) is \(-\frac{3}{2}\).
• The y-intercept \( b \) is the constant value. For \( y = -\frac{3}{2}x + \frac{3}{2} \), the y-intercept \( b \) is \(\frac{3}{2}\).

The slope \( m \) tells you how much \( y \) changes for a change in \( x \). A negative slope implies that as \( x \) increases, \( y \) decreases. The y-intercept is the point where the line crosses the y-axis, providing the starting value when \( x = 0 \).

Understanding both the slope and intercept helps predict and visualize how a line behaves on a graph.