Question

Solve the following system of linear equations by graphing describe what the graph looks like and what this tells you about the solution to the system of linear equation y=x+3 -4x+4y=28

Answer 1

To solve the system of linear equations by graphing, we'll start with the two equations you provided:

1. \( y = x + 3 \)
2. \( -4x + 4y = 28 \)

### Step 1: Simplify the second equation

We can rearrange the second equation to the slope-intercept form \( y = mx + b \). Start with:

\[
-4x + 4y = 28
\]

Add \( 4x \) to both sides:

\[
4y = 4x + 28
\]

Now, divide everything by 4:

\[
y = x + 7
\]

### Step 2: Set up the equations to graph

Now we have the two equations in slope-intercept form:

1. \( y = x + 3 \)
2. \( y = x + 7 \)

### Step 3: Graph the equations

- **Graph \( y = x + 3 \)**:
- This line has a slope of 1 and a y-intercept of 3. It passes through the point (0, 3) and rises 1 unit for each 1 unit it moves to the right.

- **Graph \( y = x + 7 \)**:
- This line also has a slope of 1 but a y-intercept of 7. It passes through the point (0, 7) and similarly rises 1 unit for each 1 unit it moves to the right.

### Step 4: Analyze the Graph

When you graph these two lines, you’ll notice:

- Both lines have the same slope (1), which means they are parallel.
- The line \( y = x + 3 \) intersects the y-axis at (0, 3), and the line \( y = x + 7 \) intersects the y-axis at (0, 7).

### Conclusion

Since the two lines are parallel, they will never intersect. This indicates that the system of linear equations has **no solution**. In other words, the equations represent two lines that will never meet. Therefore, the system is inconsistent.

### Final Statement

To summarize, the graph consists of two parallel lines, representing that the system of equations has no solutions.