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, as an 8th grader. y=x+3 -4x+4y=28

To solve the system of equations \( y = x + 3 \) and \( -4x + 4y = 28 \) by graphing, let’s first rearrange the second equation into slope-intercept form (\(y = mx + b\)) so we can easily graph both equations.

1. Start with the second equation:
\[
-4x + 4y = 28
\]

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

3. Now, divide every term by 4:
\[
y = x + 7
\]

Now we have the two equations to graph:
- \( y = x + 3 \)
- \( y = x + 7 \)

### Graphing the Equations:

1. **Graph \( y = x + 3 \)**:
- The y-intercept is \(3\) (where the line crosses the y-axis).
- You can plot another point by using \(x = 0\) to find \(y\). When \(x = 0\), \(y = 3\). This gives the point (0, 3).
- When \(x = 1\), \(y = 4\), giving the point (1, 4).
- Connect these points with a straight line.

2. **Graph \( y = x + 7 \)**:
- The y-intercept is \(7\).
- At \(x = 0\), \(y = 7\), which is the point (0, 7).
- When \(x = 1\), \(y = 8\), giving the point (1, 8).
- Connect these points with a straight line.

### Observations:

When you graph these two lines, you'll notice that they are parallel.

- The line for \(y = x + 3\) runs through (0, 3) and (1, 4).
- The line for \(y = x + 7\) runs through (0, 7) and (1, 8).

Since the lines never intersect, they don’t share any points.

### Conclusion:

This tells us that the system of equations has no solution. In other words, there isn’t a pair of values for \(x\) and \(y\) that can satisfy both equations at the same time. We say that the system is inconsistent, because the lines are parallel and will never meet. This is an important concept in algebra, showing how two relationships can have different outcomes in a consistent system, even if they both have the same slope.