Question

The system of linear equations on my graph shows two parallel lines that aren't intersecting, meaning they have no solution. Another way of showing this is by solving the two linear equations, then explaining the graph. Let's start with the equation -4x+4y=28. First, let's add -4 by 4 to get x+4y=28, then divide 4 on both sides to get x+y=7. Finally, we convert this into slope intercept form (y=mx+b) like y=x+7, where m is the slope and b is the y-intercept. Since y=x+3 is already in slope-intercept form, we can now graph these equation. When graphed, the line for y=x+3 intersects the y-axis at (0, 3), while the line for y=x+7 intersects the y-axis at (0, 7). If the slopes, m, are the same, and the y-intercepts, b, are different, there is no solution. Since the slope (1) is the same and the y-intercepts are different, the two lines are parallel, as they will never cross each other, meaning that this system of equations has no solution. In conclusion, the two lines are parallel and will never intersect, meaning it has no solution. This tells me that it's crucial to recognize those that are either one, infinite, no solution when analyzing systems of linear equations.

does this answer the problem

Solve the following system of linear equations by graphing. Graph on your own piece of paper. In your submitted answer, describe what the graph looks like and what this tells you about the solution to the system of linear equations. y=x+3 −4x+4y=28 (2 points)

Answer 1

Your explanation generally covers the problem well, but there are a few inaccuracies and clarifications needed. Let's break it down and refine the response for clarity and correctness:

1. **Rearranging the First Equation:**
You started with the equation \(-4x + 4y = 28\). When you tried to isolate \(y\), the steps were incorrectly stated. Here's the correct approach:

1. Add \(4x\) to both sides:
\[
4y = 4x + 28
\]
2. Divide by 4:
\[
y = x + 7
\]

2. **The Second Equation:**
The second equation you have is already given in slope-intercept form as \(y = x + 3\).

3. **Finding the Slope and Y-Intercept:**
- For \(y = x + 7\), the slope \(m_1\) is 1, and the y-intercept \(b_1\) is 7.
- For \(y = x + 3\), the slope \(m_2\) is also 1, and the y-intercept \(b_2\) is 3.

4. **Graph Description:**
- The graph will show two lines: one that intersects the y-axis at (0, 7) and one at (0, 3).
- Both lines have the same slope (1), which indicates they are parallel.

5. **Conclusion**:
Since the slopes are equal and the y-intercepts are different, the lines will never intersect, confirming that the system of equations has no solution.

### Revised Answer:

The system of linear equations consists of the two equations \(y = x + 3\) and \(y = x + 7\).

1. The first equation, \(y = x + 3\), intersects the y-axis at (0, 3).
2. The second equation, converted from \(-4x + 4y = 28\) to \(y = x + 7\), intersects the y-axis at (0, 7).

Both lines have the same slope of 1, indicating that they are parallel. Since the y-intercepts are different, the lines will never intersect.

### Graph Appearance:
When graphed, the line \(y = x + 3\) passes through (0,3) and the line \(y = x + 7\) passes through (0,7). This visually confirms that the lines are parallel.

### Conclusion:
Since the lines are parallel and do not cross each other at any point, this system of equations has no solution. Recognizing the relationships between the slopes and y-intercepts is essential in analyzing systems of linear equations and determining which scenarios yield unique solutions, no solutions, or infinite solutions.

This revised explanation should clearly address the problem and accurately describe the graph and its implications regarding the solutions to the equations.