Question

True or False. The lines 2 x + y = 4 and \(4 x+2 y=0\) are parallel.

Short answer

True. Both lines have the same slope of -2, so they are parallel.

Step-by-step solution

Determine the slope of the first line

Rewrite the equation of the first line, 2x + y = 4, in slope-intercept form (y = mx + b). Solve for y: y = -2x + 4.The slope (m) of the first line is -2.

Determine the slope of the second line

Rewrite the equation of the second line, 4x + 2y = 0, in slope-intercept form (y = mx + b). Solve for y: y = -2x.The slope (m) of the second line is -2.

Compare the slopes

In order to determine if the lines are parallel, compare the slopes of both lines. If the slopes are identical, the lines are parallel. Both lines have a slope of -2.

Conclusion

Since both lines have the same slope of -2, they are parallel.

Key concepts

slope-intercept form

The slope-intercept form of a linear equation is a way to express the equation of a line. This form is especially useful because it readily shows the slope of the line and where it crosses the y-axis.
The general form of a linear equation in slope-intercept form is: y = mx + b
Here:

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

For the first line, 2x + y = 4, we solved for y to convert it into the slope-intercept form and got y = -2x + 4. Here, the slope (m) is -2, and the y-intercept (b) is 4.
Similarly, for the second line, 4x + 2y = 0, we rewrote it in slope-intercept form as y = -2x. In this case, the slope (m) is -2, and the y-intercept (b) is 0.
This form is easy to use and helps in quickly identifying the properties of the line.

slope calculation

The slope of a line measures its steepness and direction. It can be calculated when the line is expressed in slope-intercept form, as seen previously.
To find the slope, you need to isolate the variable y and get the equation into the form y = mx + b. The coefficient of x in this equation is the slope (m).
For the line 2x + y = 4:

• Subtract 2x from both sides: y = -2x + 4
• The slope (m) is -2

For the line 4x + 2y = 0:

• Subtract 4x from both sides: 2y = -4x
• Divide by 2: y = -2x
• The slope (m) is -2

With these steps, you can easily find the slope of any line when given its equation. Just follow the process of isolating y and identifying the coefficient of x.

linear equations

Linear equations are equations that graph as straight lines on the Cartesian plane. They can be written in multiple forms, but the most commonly used is the slope-intercept form, as discussed earlier.
Linear equations can generally be expressed as: Ax + By = C
Here:

• A, B, and C are constants
• x and y are variables

To determine whether two lines are parallel, their slopes must be compared. In the exercise, we had two linear equations: 2x + y = 4 and 4x + 2y = 0.
By converting them to the slope-intercept form, we could easily identify the slopes of both lines, which turned out to be identical (-2 for both). Therefore, the lines are parallel as they have the same slope but different y-intercepts.