Question

Place an \(X\) next to the two of the four lines below that definitely form part of a parallelogram. Line A: \(3 x-2 y=10\) Line B: \(y=-\frac{2}{3} x-5\) Line C: \(3 x+2 y=10\) Line D: \(y=\frac{3}{2} x-15\)

Short answer

Line A and Line D. Line B and Line C.

Step-by-step solution

- Find Slopes of Lines

First, find the slopes of each line. Rewrite each equation in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope.

Step 1a - Find Slope of Line A

Line A: \(3x - 2y = 10\). Rewrite as \(y = \frac{3}{2} x - 5\). The slope of Line A is \(\frac{3}{2}\).

Step 1b - Find Slope of Line B

Line B: \(y = -\frac{2}{3} x - 5\). The slope is \(-\frac{2}{3}\) by inspection.

Step 1c - Find Slope of Line C

Line C: \(3x + 2y = 10\). Rewrite as \(y = -\frac{3}{2} x + 5\). The slope of Line C is \(-\frac{3}{2}\).

Step 1d - Find Slope of Line D

Line D: \(y = \frac{3}{2} x - 15\). The slope is \(\frac{3}{2}\) by inspection.

- Identify Parallel Lines

In a parallelogram, opposite sides are parallel. Lines with the same slopes are parallel.

Step 2a - Check Parallel Lines

Line A and Line D both have a slope of \(\frac{3}{2}\), so they are parallel. Line B and Line C both have a slope of \(-\frac{3}{2}\), so they are also parallel.

Key concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is very useful in geometry and algebra. This form is written as
\text{y = mx + b}
where \( m \) is the slope and \( b \) is the y-intercept.
The slope \( m \) shows how steep the line is and whether it rises or falls.
When solving for the slope-intercept form from a standard form equation like \(3x - 2y = 10\), you need to isolate \( y \) on one side.
Rewrite the equation:
\(3x - 2y = 10\)
Subtract \( 3x \) from both sides: \( -2y = -3x + 10 \)
Divide every term by \( -2 \): \( y = \frac{3}{2} x - 5 \)
Now, it is in the slope-intercept form with \( m = \frac{3}{2} \) and \( b = -5 \).
This form simplifies identifying the slope and y-intercept, allowing us to quickly understand the line's behavior.

Parallel Lines

Parallel lines are lines in a plane that do not meet; they have the same slope but different y-intercepts. To identify parallel lines, compare the slopes.

• Line A \( 3x - 2y = 10 \) has a slope of \( \frac{3}{2} \).


• Line B \( y = -\frac{2}{3} x - 5 \) has a slope of \( -\frac{2}{3} \).


• Line C \( 3x + 2y = 10 \) has a slope of \( -\frac{3}{2} \).


• Line D \( y = \frac{3}{2} x - 15 \) has a slope of \( \frac{3}{2} \).

We can see that:

• Line A and Line D are parallel because they both have a slope of \( \frac{3}{2} \).


• Line B and Line C are parallel because they both have a slope of \( -\frac{3}{2} \).

Parallel lines are fundamental when identifying shapes like parallelograms.

Parallelogram Properties

A parallelogram has some distinct properties based on its sides, angles, and diagonals.

• Opposite sides are parallel and equal in length.


• Opposite angles are equal.


• The diagonals bisect each other.

For example, in our exercise, if two pairs of lines are identified as parallel, those lines can form a parallelogram.
For Lines A \( 3x - 2y = 10 \) and D \( y = \frac{3}{2} x - 15 \), they both have the same slope of \( \frac{3}{2} \), making them parallel. Similarly, Lines B \( y = -\frac{2}{3} x - 5 \) and C \( 3x + 2y = 10 \) share a slope of \( -\frac{2}{3} \), making them parallel and fulfilling a key property for forming a parallelogram.

GED Math

Understanding geometry and algebra concepts are crucial for the GED math test. You will encounter problems involving slopes, lines, shapes, and properties of figures.
Key concepts to focus on include:

• Slope-Intercept Form: Ability to convert equations and identify slopes and intercepts.


• Parallel Lines: Recognizing and verifying parallel lines using slopes.


• Parallelogram Properties: Identifying properties that define parallelograms and other shapes.

In the given problem, applying these principles helps to determine which lines form a parallelogram, ensuring a comprehensive understanding needed for the GED math test.