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 solve the system of linear equations by graphing, we need to first rewrite the equations in slope-intercept form (y = mx + b).
1) y = x + 3
To rewrite this equation in slope-intercept form, we can see that the coefficient of x is 1, thus the slope is 1. The y-intercept is 3. Therefore, the equation becomes:
y = 1x + 3
2) -4x + 4y = 28
To make this equation easier to work with, we can divide both sides by 4, resulting in:
-x + y = 7
Rearranging gives us:
y = x + 7
Now that we have both equations in slope-intercept form, we can graph them.
For the first equation, y = x + 3, we can start by plotting the y-intercept at (0, 3). From there, we can use the slope of 1 (rise of 1, run of 1) to find another point. For example, if we move 1 unit to the right and 1 unit up from the y-intercept, we get the point (1, 4). Plotting these points and drawing a straight line through them, the graph of the first equation would look like a diagonal line slanting upwards.
For the second equation, y = x + 7, we once again plot the y-intercept at (0, 7) and use the slope of 1 to find another point. Moving 1 unit to the right and 1 unit up from the y-intercept, we get the point (1, 8). Plotting these points and drawing a straight line through them, the graph of the second equation would also look like a diagonal line slanting upwards.
Now that we have both graphs, we can see that they intersect at a single point. This tells us that the system of linear equations has a unique solution. To find the coordinates of the intersection point, we can solve the system of equations algebraically or read the approximate values from the graph.