Systems of Linear Equations Unit Test

Question

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Question
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.

y=x+3


−4x+4y=28


(2 points)

Answer 1

To solve the given system of linear equations by graphing, you need to graph both equations and analyze their intersection.

Step 1: Convert the second equation to slope-intercept form

The second equation is \(-4x + 4y = 28\). First, we can solve for \(y\):

1. Add \(4x\) to both sides: \[ 4y = 4x + 28 \]

2. Divide everything by 4: \[ y = x + 7 \]

Step 2: Graph the equations

Now we can graph both equations:

1. For the first equation \(y = x + 3\):

   • The y-intercept is 3, so plot the point (0,3).
   • The slope is 1, meaning that for every 1 unit you move right, you move up 1 unit. You can plot another point at (1,4) (moving one unit right and one unit up from (0,3)).
   • Draw a line through these points.

2. For the second equation \(y = x + 7\):

   • The y-intercept is 7, so plot the point (0,7).
   • This line will have the same slope of 1. The next point can be plotted at (1,8).
   • Draw a line through these points.

Step 3: Analyze the graph

When you graph these two lines, you will see that they are parallel because they have the same slope (slope = 1) but different y-intercepts (3 and 7). Since parallel lines never intersect, this means that there is no solution to the system of equations.

Conclusion

The graph consists of two parallel lines that do not meet. This indicates that the system of linear equations has no solution.