Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(57-64\), write an equation in the form \(y=m x+b\) of the line that is described. The \(y\) -intercept is 5 and the line is parallel to the line whose equation is \(3 x+y=6\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -3x + 5\)

Step 1: Find the slope of the given line

First, rearrange the equation \(3x + y = 6\) into slope-intercept form, by subtracting \(3x\) from both sides to get \(y = -3x + 6\). From this equation it can be seen that, the slope \(m\) of the given line is -3.

Step 2: Use the property of parallel lines

Parallel lines have the same slope. Hence, the slope of the line we are looking for is also -3.

Step 3: Use the slope-intercept form

Supplemented by the given y-intercept 5, we can apply these values to the slope-intercept formula \(y = mx + b\) and hence the equation of the line is \(y = -3x + 5\)

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a popular way to describe a straight line. This form is denoted as \( y = mx + b \), where:

• \( y \) represents the dependent variable.
• \( m \) is the slope of the line.
• \( x \) is the independent variable.
• \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

This form is particularly useful because it instantly reveals both the slope and the y-intercept of the line.
In the context of solving linear equations or drawing graphs, having the equation in this format makes it easy to gather important information quickly and efficiently.
For example, given the slope-intercept form \( y = -3x + 5 \), we can quickly identify that the slope is -3, and the line crosses the y-axis at 5.

Parallel Lines

Parallel lines are lines in a plane that never meet; they are always the same distance apart and have the same slope.
When thinking about parallel lines in relation to linear equations, it is important to remember:

• Parallel lines have identical slopes.
• They never intersect each other regardless of how far extended.

For instance, if one line has an equation of \( y = -3x + 6 \), another line with a slope of -3, such as \( y = -3x + 5 \), would be parallel to it.
Despite their different y-intercepts, the parallel nature is maintained because their slopes are equal. This property is vital in geometry and algebra for identifying lines that never cross.

Slope Calculation

Calculating the slope of a line is a fundamental concept in understanding linear relationships. The slope is a measure of the steepness or incline of the line and is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate between two distinct points on the line.

• Mathematically, slope \( m \) is given by the formula: \( m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \).
• A positive slope means the line goes upward as you move from left to right, while a negative slope indicates the line goes downward.
• A zero slope signifies a horizontal line, and undefined slope corresponds to a vertical line.

For the line given by \( 3x + y = 6 \), by rearranging it into slope-intercept form, \( y = -3x + 6 \), we can directly see that the slope \( m \) is -3.
This process shows that linear equations can often be transformed to quickly reveal key properties of the line, such as its slope.