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
To graph the first equation, y = x + 3, we can start by plotting a few points. We can choose any values for x and then find the corresponding y values.
Let's choose x = 0. When x = 0, y = 0 + 3 = 3. So we have the point (0, 3).
Next, let's choose x = 1. When x = 1, y = 1 + 3 = 4. So we have the point (1, 4).
Lastly, let's choose x = -1. When x = -1, y = -1 + 3 = 2. So we have the point (-1, 2).
Now, let's plot these points on a graph and draw a line through them.
The graph of y = x + 3 is a straight line that passes through the points (0, 3), (1, 4), and (-1, 2). It has a positive slope, which means that the line is slanting upwards from left to right.
Now let's graph the second equation, -4x + 4y = 28. We can rewrite this equation in slope-intercept form (y = mx + b) by isolating y:
-4x + 4y = 28
4y = 4x + 28
y = x + 7
Using the same method as before, we can plot a few points for the second equation.
When x = 0, y = 0 + 7 = 7. So we have the point (0, 7).
When x = 1, y = 1 + 7 = 8. So we have the point (1, 8).
When x = -1, y = -1 + 7 = 6. So we have the point (-1, 6).
Plotting these points on the graph, we see that the graph of y = x + 7 is a straight line passing through the points (0, 7), (1, 8), and (-1, 6). This line also has a positive slope, slanting upwards from left to right.
Now, let's look at the two graphs together. The two lines intersect at a single point, which is (4, 7). This is the solution to the system of linear equations.
The fact that the lines intersect at a single point tells us that there is a unique solution for the system of linear equations. The coordinates of this point represent the values of x and y that satisfy both equations simultaneously.