Problem 4 - Precalculus: Mathematics for Calculus

Question

For the linear equation \(2 x+3 y-12=0,\) the \(x\)-intercept is __________ and the \(y\)-intercept is _________. The equation in slope-intercept form is \(y=\) ____________. The slope of the graph of this equation is ____________.

Step-by-Step Solution

Answer

The x-intercept is 6; The y-intercept is 4; Equation: \( y = -\frac{2}{3}x + 4 \); Slope: \( -\frac{2}{3} \).

Step 1: Finding the x-intercept

The x-intercept is the point where the graph of the equation crosses the x-axis. To find this, set \( y = 0 \) in the equation and solve for \( x \). Start with the given equation: \[ 2x + 3y - 12 = 0 \] Setting \( y = 0 \) gives: \[ 2x - 12 = 0 \] Solving for \( x \), we add 12 to both sides: \[ 2x = 12 \] Divide both sides by 2: \[ x = 6 \] Thus, the x-intercept is \( 6 \).

Step 2: Finding the y-intercept

The y-intercept is the point where the graph of the equation crosses the y-axis. To find this, set \( x = 0 \) in the equation and solve for \( y \). Start with the given equation: \[ 2x + 3y - 12 = 0 \] Setting \( x = 0 \) gives: \[ 3y - 12 = 0 \] Solving for \( y \), we add 12 to both sides: \[ 3y = 12 \] Divide both sides by 3: \[ y = 4 \] Thus, the y-intercept is \( 4 \).

Step 3: Converting to slope-intercept form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) represents the slope, and \( b \) represents the y-intercept. Start from the equation: \[ 2x + 3y - 12 = 0 \] We need to solve for \( y \). Subtract \( 2x \) from both sides: \[ 3y = -2x + 12 \] Divide everything by 3 to isolate \( y \): \[ y = -\frac{2}{3}x + 4 \] This is the equation in slope-intercept form.

Step 4: Finding the slope of the line

The slope of a line in slope-intercept form \( y = mx + b \) is given by \( m \). From the slope-intercept form of the equation obtained previously: \[ y = -\frac{2}{3}x + 4 \] The slope \( m \) is \(-\frac{2}{3}\).

Key Concepts

Intercepts

When working with linear equations, understanding intercepts is crucial. Intercepts are points where the graph of a line crosses the axes on a coordinate plane. There are two types of intercepts:

• **x-intercept:** This is the point where the line crosses the x-axis. To find it, we set the value of y to zero in the equation and solve for x. For the equation given: \[ 2x + 3(0) - 12 = 0 \] Solving it gives us: \[ x = 6 \] So, the x-intercept is 6.


• **y-intercept:** This is the point where the line cuts through the y-axis. We find it by setting x to zero and solving for y in the equation: \[ 2(0) + 3y - 12 = 0 \] After solving, we find: \[ y = 4 \] Therefore, the y-intercept is 4.

The intercepts give you key points that help in sketching the graph of the line quickly.

Slope-Intercept Form

The slope-intercept form is a common way to express the equation of a straight line. It is written as: \[ y = mx + b \] Here, \(m\) stands for the slope of the line and \(b\) is the y-intercept. This form is incredibly useful because it directly shows the slope and the y-intercept, helping to quickly analyze the line's behavior.
To convert a linear equation to slope-intercept form, you'll need to rearrange the original equation so that y is isolated on one side. For our example: \[ 2x + 3y - 12 = 0 \]Rearrange it to get:\[ 3y = -2x + 12 \]Divide everything by 3:\[ y = -\frac{2}{3}x + 4 \]Now, you have it in slope-intercept form. This makes it easy to identify the slope as \(-\frac{2}{3}\) and the y-intercept as 4.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It is often represented by the letter \(m\). Slope can be determined when a line is in the slope-intercept form, \[ y = mx + b \]It quantifies how much y increases or decreases as x increases by one unit. The slope of a line can reveal a lot about the line's direction:

• **Positive Slope:** Means the line is increasing or going upwards as you move from left to right.


• **Negative Slope:** Indicates that the line is descending as you go from left to right.

For our specific equation: \[ y = -\frac{2}{3}x + 4 \]The slope \(m\) is \(-\frac{2}{3}\). This negative slope tells us that the line decreases in y by 2 units for every 3 units increase in x. Understanding the slope helps predict how the line will look on the graph.