solve the system of equation by graphing then classify the system as consistent or inconsistent and as dependent or independent
8x-4y=-20
4y-8x=-20
To solve the system of equations by graphing, follow these steps:
Step 1: Write down the equations of the system.
The given system of equations is:
8x - 4y = -20
4y - 8x = -20
Step 2: Convert the equations into slope-intercept form (y = mx + b), if possible.
Let's convert the first equation:
8x - 4y = -20
-4y = -8x - 20 (subtracted 8x from both sides)
y = 2x + 5 (divided all terms by -4)
Now let's convert the second equation:
4y - 8x = -20
4y = 8x - 20 (added 8x to both sides)
y = 2x - 5 (divided all terms by 4)
Step 3: Graph the equations on the same coordinate plane.
To graph the equations, create a table of values for both equations and plot the points on the graph. Since both equations are already in slope-intercept form (y = mx + b), you can easily find points by selecting values of x and then calculating the corresponding y values.
For the first equation (y = 2x + 5):
Let x = 0, then y = 2(0) + 5 = 5. So we have the point (0, 5).
Let x = 1, then y = 2(1) + 5 = 7. So we have the point (1, 7).
Let x = -1, then y = 2(-1) + 5 = 3. So we have the point (-1, 3).
Plot these points on the graph.
For the second equation (y = 2x - 5):
Let x = 0, then y = 2(0) - 5 = -5. So we have the point (0, -5).
Let x = 1, then y = 2(1) - 5 = -3. So we have the point (1, -3).
Let x = -1, then y = 2(-1) - 5 = -7. So we have the point (-1, -7).
Plot these points on the graph.
Step 4: Interpret the graph.
After plotting the equations on the same graph, observe the points. If the lines intersect at a single point, then the system has a unique solution and is considered consistent and independent. If the lines are coincident (overlapping) or parallel, there is no intersection point, and the system is either dependent or inconsistent.
In this case, you will notice that the two lines have the same slope (both equations have a slope of 2) and different y-intercepts. The lines are coincident (overlapping), meaning they coincide with each other and have infinite solutions. Therefore, the system is consistent and dependent.
To summarize:
- The system of equations is consistent because it has at least one solution.
- The system of equations is dependent because the equations are coincident (overlapping).