To solve the system of linear equations by graphing, we start by rewriting each equation in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
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First equation: \(y = x + 3\)
This equation is already in slope-intercept form. The slope (m) is 1, and the y-intercept (b) is 3. This means that we start at the point (0, 3) on the y-axis and rise 1 unit up and 1 unit to the right to find another point (1, 4). We can use this to draw the line.
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Second equation: \(-4x + 4y = 28\)
To put this in slope-intercept form, we first isolate \(y\): \[ 4y = 4x + 28 \] \[ y = x + 7 \]
Now, this equation is also in slope-intercept form. The slope here is also 1, and the y-intercept is 7. So, we start at (0, 7) on the y-axis and rise 1 unit and run 1 unit to the right to find another point (1, 8).
Now that we have both equations in slope-intercept form, we can graph them:
- For \(y = x + 3\), plot the points (0, 3) and (1, 4), and draw a straight line through those points.
- For \(y = x + 7\), start at (0, 7) and plot (1, 8) and draw a line through those points.
Graph Description:
The graph has two straight lines, both with a slope of 1. The first line (from \(y = x + 3\)) passes through points (0, 3) and (1, 4). The second line (from \(y = x + 7\)) goes through points (0, 7) and (1, 8).
Upon drawing these lines, you will notice they are parallel to each other because they have the same slope but different y-intercepts (3 and 7).
Conclusion:
Since the lines are parallel, this tells us that there is no point where they intersect. As a result, the system of equations has no solution. In other words, the equations represent two different lines that will never meet on a graph.
This conclusion makes sense because if two lines are parallel, it means that there are no values of \(x\) and \(y\) that can satisfy both equations at the same time.
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