Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(35-38\), find the slope and \(y\) -intercept of each line. \(6-2 y=10 x-2\)

Step-by-Step Solution

Answer

The slope is 5 and the y-intercept is -2.

Step 1: Rearrange the Equation

We start with the equation \(6 - 2y = 10x - 2\). To make it easier to find the slope and y-intercept, we need to rearrange it into the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Move Terms Involving y

Add \(2y\) to both sides of the equation to get rid of the negative sign in front of \(y\). This gives us: \(6 = 10x - 2 + 2y\).

Step 3: Isolate the y-Term

Subtract \(10x\) and add \(2\) to both sides of the equation to isolate the \(y\) term. This results in: \(2y = 10x - 4\).

Step 4: Solve for y

Divide every term in the equation by \(2\) to solve for \(y\). We get: \(y = 5x - 2\).

Step 5: Identify Slope and y-Intercept

Now that the equation is in the form \(y = 5x - 2\), we can identify the slope \(m\) as \(5\) and the y-intercept \(b\) as \(-2\).

Key Concepts

Slope

The concept of slope is central to understanding linear equations. Simply put, the slope of a line measures how steep the line is. It is calculated as the ratio of the rise (vertical change) to the run (horizontal change).
For any two points on a line,

• if you move from one point to another, the slope tells you how much up or down (rise) you move for a certain amount of right or left (run).
• If the slope is positive, the line rises as it moves from left to right.
• If it is negative, the line falls as it moves from left to right.

In the slope-intercept form of a linear equation, which is written as \(y = mx + b\), the slope is represented by \(m\). Finding the slope from an equation in this form is straightforward, as it is simply the coefficient of \(x\).
For example, in the equation \(y = 5x - 2\), the slope \(m\) is 5.

y-Intercept

The y-intercept is the point at which a line crosses the y-axis. This happens when the value of \(x\) is zero.
In the slope-intercept form \(y = mx + b\), the \(y\)-intercept is represented by the constant term \(b\). This means that when \(x = 0\), the value of \(y\) is equal to \(b\).

• The y-intercept can tell you a lot about the line. It helps in setting the starting point of the line on the graph.
• It is especially useful when sketching a graph from an equation.

In our example \(y = 5x - 2\), the y-intercept \(b\) is \(-2\). So the line crosses the y-axis at \(y = -2\).

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of writing linear equations so that the slope and y-intercept are immediately visible. This form is especially useful for graphing equations quickly.
The general formula is \(y = mx + b\), where:

• \(y\) is the dependent variable or output.
• \(x\) is the independent variable or input.
• \(m\) is the slope, indicating how steep the line is.
• \(b\) is the y-intercept, showing where the line crosses the y-axis.

This form is convenient because it allows you to easily draw the graph by starting at the y-intercept \(b\) and using the slope \(m\) to find other points on the line. For the equation \(y = 5x - 2\), you know the line crosses through (0, -2) and rises 5 units for every unit it runs.

Equation Rearrangement

Sometimes, linear equations are not presented in the slope-intercept form, and you need to rearrange them, so important features like slope and y-intercept are highlighted.
Let's take a look at how you can rearrange any linear equation to the form \(y = mx + b\).

• First, you should aim to isolate the \(y\)-term on one side of the equation.
• You may need to move terms around by adding or subtracting them from both sides.
• Then, simplify by dividing or multiplying both sides of the equation to make \(y\) stand alone.

For example, starting with \(6 - 2y = 10x - 2\), we can rearrange it:

• Add \(2y\) to both sides to keep \(y\) positive.
• Next, isolate the \(y\)-term, resulting in \(2y = 10x - 4\).
• Finally, divide every term by \(2\). This simplifies to \(y = 5x - 2\).

Rearrangement is a key skill in algebra that allows you to transform equations to a form that is more usable for your needs, like graphing or identifying key features.