To solve the system of linear equations by graphing, we will first rearrange both equations into a form that is easy to graph.
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The first equation is already in slope-intercept form (y = mx + b): \[ y = x + 3 \] This equation has a slope of 1 and a y-intercept at (0, 3). This means you can start at the point (0, 3) on the graph, and from there move up 1 unit and right 1 unit to plot another point.
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Now let's rearrange the second equation: \[ -4x + 4y = 28 \] Divide everything by 4 for simplicity: \[ -x + y = 7 \] Rearranging gives: \[ y = x + 7 \] This equation also has a slope of 1, but a y-intercept of (0, 7). Similarly, you can use the y-intercept to plot the graph.
Graph Description
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Graph of the First Equation: Plotting \( y = x + 3 \), you start at (0, 3) and move to (1, 4), (2, 5), and so on. This line will have points like (0, 3), (1, 4), (2, 5), and will continue with a slope of 1 in a diagonal line.
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Graph of the Second Equation: For \( y = x + 7 \), you start at (0, 7) and move to (1, 8), (2, 9), etc. This line will contain points like (0, 7), (1, 8), (2, 9), also forming a diagonal line but higher up on the graph.
Intersection Point
When you graph both lines, you will find that they are parallel. Since both lines have the same slope (1) but different y-intercepts (3 and 7), they will never intersect.
Conclusion
The graph visually shows that there are no points at which the two lines intersect, which means that the system of linear equations has no solution. This tells you that the two equations represent parallel lines in a two-dimensional space, confirming that they cannot satisfy both equations simultaneously.