To solve the system of linear equations by graphing, we'll start with the two equations you provided:
- \( y = x + 3 \)
- \( -4x + 4y = 28 \)
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:
- \( y = x + 3 \)
- \( y = x + 7 \)
Step 3: Graph the equations
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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.
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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.