https://file.aiquickdraw.com/mathbot/file/172555107072795sdpo4p.jpg

The image shows two equations:

1. \(-3x + 4y = 4\)
2. \(x - \frac{4}{3}y = -\frac{4}{3}\)

We are tasked with determining which of the following statements is true about these two equations:

A. They have the same y-intercept.
B. They are parallel lines.
C. They have the same x-intercept.
D. They are perpendicular lines.
E. They are the same line.

### Step 1: Rewrite both equations in slope-intercept form (\(y = mx + b\)) to compare them.

#### First equation: \(-3x + 4y = 4\)
Solve for \(y\):
\[
4y = 3x + 4
\]
\[
y = \frac{3}{4}x + 1
\]
This equation has a slope of \(\frac{3}{4}\) and a y-intercept of 1.

#### Second equation: \(x - \frac{4}{3}y = -\frac{4}{3}\)
Solve for \(y\):
\[
-\frac{4}{3}y = -x - \frac{4}{3}
\]
Multiply through by \(-3/4\) to isolate \(y\):
\[
y = \frac{3}{4}x + 1
\]
This equation also has a slope of \(\frac{3}{4}\) and a y-intercept of 1.

### Step 2: Analyze the relationship between the two lines.
Both equations have the same slope \(\frac{3}{4}\) and the same y-intercept 1. Therefore, these two lines are identical.

### Conclusion:
The correct answer is **E. They are the same line**.

Would you like any further details on the steps, or do you have any questions?

### Here are 5 related questions to explore this topic further:
1. How do you determine if two lines are parallel based on their equations?
2. What is the significance of the slope in a linear equation?
3. How can you find the point of intersection between two lines?
4. What happens when two lines have the same slope but different y-intercepts?
5. Can two lines be both parallel and perpendicular under any conditions?

### Tip:
When comparing two lines, always convert the equations into slope-intercept form (\(y = mx + b\)) to easily compare their slopes and intercepts.

Determine what is true about each pair of equations:

-3x + 4y = 4 and x - (4/3)y = -4/3

Analyze and compare two linear equations -3x + 4y = 4 and x - (4/3)y = -4/3. Learn how to identify if they are the same line, parallel, or perpendicular by converting them into slope-intercept form.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation