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Solve the following system of linear equations by graphing. Graph on your own piece of paper. In your submitted answer, describe what the graph looks like and what this tells you about the solution to the system of linear equations, as an 8th grader.

y=x+3

−4x+4y=28

3 answers

To solve the system of linear equations by graphing, we start with the two equations:

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

Step 1: Graph the first equation

The first equation, \( y = x + 3 \), is in slope-intercept form (\( y = mx + b \)), where \( m = 1 \) (the slope) and \( b = 3 \) (the y-intercept).

  • To graph this line:
    • The y-intercept is at (0, 3). This is the point where the line crosses the y-axis.
    • From this point, you can use the slope to find another point. Since the slope is 1, you go up 1 unit and right 1 unit. This leads you to the point (1, 4).
    • Connect these points with a straight line.

Step 2: Graph the second equation

Now, we need to rewrite the second equation \( -4x + 4y = 28 \) in slope-intercept form.

  • Start by isolating \( y \): \[ 4y = 4x + 28 \] \[ y = x + 7 \]

Now we have the second equation in slope-intercept form, where the slope \( m = 1 \) and the y-intercept \( b = 7 \).

  • To graph this line:
    • The y-intercept is at (0, 7).
    • From that point, use the slope (1) to find another point. Go up 1 unit and right 1 unit to get to (1, 8).
    • Connect these points with a straight line.

Step 3: Analyze the graphs

When you plot both lines on the same graph:

  • The first line (from \( y = x + 3 \)) will cross the y-axis at (0, 3).
  • The second line (from \( y = x + 7 \)) will cross the y-axis at (0, 7).

Step 4: Identify intersections and solutions

  • Notice how the two lines are parallel because they have the same slope (1), but different y-intercepts.
  • Since parallel lines never intersect, this tells us that there is no solution to this system of equations.

Conclusion

In summary, the graph will show two parallel lines, one crossing the y-axis at (0, 3) and the other at (0, 7). This means there is no point where the two lines meet, which indicates that the system of equations has no solution.

NOW MAKE IT VERY VERY SHORT

To solve the system of equations:

  1. \( y = x + 3 \)
  2. \( -4x + 4y = 28 \) (rewritten as \( y = x + 7 \))

Graphing:

  • The first line crosses the y-axis at (0, 3).
  • The second line crosses the y-axis at (0, 7).

Conclusion:

The lines are parallel (same slope, different intercepts), meaning they never intersect. Therefore, the system has no solution.