Problem 84 - Intermediate Algebra

Question

Are the lines described by \(y=2 x-7\) and \(2 x-y=10\) parallel?

Step-by-Step Solution

Answer

Yes, the lines are parallel because they have the same slope and different y-intercepts.

Step 1: Identify the Equation Forms

First, we need to understand what form these equations are in. The first equation is already in the slope-intercept form, which is given by \(y=mx+b\), where \(m\) is the slope. So for \(y = 2x - 7\), the slope \(m\) is 2.

Step 2: Convert Second Equation to Slope-Intercept Form

The second equation \(2x - y = 10\) is not in slope-intercept form. We convert it by solving for \(y\). Begin by subtracting \(2x\) from both sides to get \(-y = -2x + 10\). Then, multiply by \(-1\) to solve for \(y\), yielding \(y = 2x - 10\). Now, the slope \(m\) for this line is also 2.

Step 3: Compare the Slopes

Now that both equations are in slope-intercept form, compare their slopes. The first line has a slope of 2, and the second line also has a slope of 2.

Step 4: Determine If the Lines Are Parallel

Two lines are parallel if they have the same slope and different y-intercepts. We verified that both lines have the same slope of 2. Check the y-intercepts: \(-7\) for the first line and \(-10\) for the second line. Since the y-intercepts are different, the lines are indeed parallel.

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. It is given by \(y = mx + b\), where:

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

This format is particularly useful because it immediately tells you the slope and the y-intercept of a line. Knowing the slope helps in understanding the steepness and direction of the line. If the slope \(m\) is positive, the line ascends from left to right. If \(m\) is negative, the line descends.
In our problem, the equation \(y = 2x - 7\) is already in slope-intercept form, with a slope \(m = 2\) and a y-intercept \(b = -7\). This clarity allows us to quickly see these crucial elements without further calculation.

Solving Equations to Convert Forms

To work with equations, especially when comparing slopes, converting to the slope-intercept form is key. For the equation \(2x - y = 10\), it is not in slope-intercept form initially.
We aim to solve for \(y\) by rearranging the equation:

• Start with the original equation: \(2x - y = 10\).
• Subtract \(2x\) from both sides to isolate \(y\): \(-y = -2x + 10\).
• Multiply everything by \(-1\) to solve for \(y\): \(y = 2x - 10\).

Now, it is clear this line has a slope \(m = 2\). Converting the equation to this form is essential because it simplifies the comparison of different equations through their slopes and y-intercepts. This transformation shows that both lines in our exercise have the same slope, which is a key step in determining parallelism.

Understanding the Y-Intercept

The y-intercept \(b\) is where a line crosses the y-axis. This point is crucial because it uniquely defines a line in relation to the y-axis, apart from its slope. Given the equation in slope-intercept form \(y = mx + b\), you can find the y-intercept as the value of \(b\).

• For \(y = 2x - 7\), the y-intercept is \(-7\).
• For \(y = 2x - 10\), the y-intercept is \(-10\).

In our problem, to determine if the lines are parallel, we need identical slopes and different y-intercepts. The differing y-intercepts confirm that the lines, though having the same slope, do not coincide and are indeed parallel.
This distinction is essential because if lines have both the same slope and the same y-intercept, they would be the same line rather than parallel. Hence, identifying the y-intercept adds an important layer to understanding the position of a line.