solve the following system of linear equations by graphing. Graph on your own piece of paper. In your summitted answer, describe what the graph looks like and what this tells you about the solution to the system of linear equations.

Question

y=x+3

-4x+4y=28

Answer 1

To solve the system of linear equations by graphing, we first need to express both equations in a suitable form.

1. The first equation is already in slope-intercept form (y = mx + b): \[ y = x + 3 \]

2. The second equation, \(-4x + 4y = 28\), can be rearranged to slope-intercept form as follows: \[ 4y = 4x + 28 \ y = x + 7 \]

Now we have the following equations to graph:

• \(y = x + 3\)
• \(y = x + 7\)

Graphing:

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

   • The y-intercept is 3, so we plot the point (0, 3).
   • The slope is 1, meaning for every 1 unit increase in x, y increases by 1. Thus, from (0, 3), if we go right to (1, 4), that’s another point on the line.
   • Connecting these points, we draw a straight line.

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

   • The y-intercept is 7, so we plot the point (0, 7).
   • The slope is also 1, so from (0, 7), we can plot another point by going right to (1, 8).
   • Connecting these points, we draw a straight line.

Observations from the Graph:

When you graph both equations, you’ll notice that the lines are parallel. The line \(y = x + 3\) runs through the points (0, 3) and (1, 4), while the line \(y = x + 7\) runs through (0, 7) and (1, 8). Because the slopes of both lines are the same (both are 1), they will never intersect.

Conclusion:

Since the two lines are parallel, this tells us that there is no solution to the system of linear equations. In other words, the equations represent two distinct lines that do not have any points in common. Therefore, the system is inconsistent.