To solve the system of linear equations by graphing, we first need to rewrite the equations in a suitable form.
Given equations:
- \( y = x + 3 \)
- \( -4x + 4y = 28 \)
Step 1: Rearranging the second equation
We can rewrite the second equation in slope-intercept form (y = mx + b):
\[ -4x + 4y = 28 \] Divide the entire equation by 4:
\[ -y + x = 7 \] Now, isolate y:
\[ y = x - 7 \]
The equations now are:
- \( y = x + 3 \)
- \( y = x - 7 \)
Step 2: Graphing the equations
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Graph \( y = x + 3 \):
- The y-intercept is (0, 3).
- The slope is 1 (meaning for every unit increase in x, y increases by 1).
- Points on this line include (0, 3), (1, 4), (2, 5), etc.
-
Graph \( y = x - 7 \):
- The y-intercept is (0, -7).
- The slope is also 1 (meaning the line rises 1 unit for every unit increase in x).
- Points on this line include (0, -7), (1, -6), (2, -5), etc.
Step 3: Analyzing the graph
When plotted, both lines will have the same slope of 1 but different y-intercepts. This means that they will be parallel to one another.
Conclusion:
Since the lines are parallel and will never intersect, this indicates that there is no solution to the system of equations. In other words, the equations represent two parallel lines that have different outputs for the same input values.
Summary of the graph:
- The line for \( y = x + 3 \) passes through (0, 3) and rises to the right.
- The line for \( y = x - 7 \) passes through (0, -7) and also rises to the right.
- The absence of any intersection point between these two lines signifies that the system of equations has no solution.