Solve the following system of equations by graphing. If the system is inconsistent or the equations are dependent, say so.
X-y=2
X+y=6
X-y=2
X+y=6
------------------- add
2 X = 8
so
X = 4
then
y = 6-4 = 2
or
y = 4-2 = 2
To solve the system of equations by graphing, we will plot the lines represented by each equation on a graph and find their point of intersection, if any.
The first equation is X - y = 2.
Rearranging this equation, we have y = x - 2.
The second equation is X + y = 6.
Rearranging this equation, we have y = 6 - x.
Now, let's plot these lines on a graph:
Graph of y = x - 2:
^
|
* |
|
* |
|
* |
|
------------------------>
|-----|
2
Graph of y = 6 - x:
^
|
* |
|
* |
|
* |
|
------------------------>
|-----|
6
From the graph, we can see that the lines intersect at the point (4, 2).
Therefore, the solution to the system of equations is x = 4 and y = 2.
The system of equations is consistent, and the equations are independent since they intersect at a unique point.
To solve the system of equations by graphing, we need to plot the equations on a coordinate plane and find the point of intersection.
For the first equation, X - y = 2, let's rearrange it to solve for y:
y = X - 2.
For the second equation, X + y = 6, rearrange it to solve for y:
y = 6 - X.
Now we can proceed to graph both equations on the same coordinate plane.
Step 1: Create a grid or coordinate plane with the X-axis and y-axis labeled.
Step 2: For the first equation, X - y = 2, pick a few X-values and find their corresponding y-values. For example:
- If X = 0, y = 0 - 2 = -2.
- If X = 2, y = 2 - 2 = 0.
- If X = 4, y = 4 - 2 = 2.
Plot these points on the coordinate plane and draw a line through them. This line represents the first equation.
Step 3: For the second equation, X + y = 6, pick a few X-values and find their corresponding y-values. For example:
- If X = 0, y = 6 - 0 = 6.
- If X = 2, y = 6 - 2 = 4.
- If X = 4, y = 6 - 4 = 2.
Plot these points on the coordinate plane and draw a line through them. This line represents the second equation.
Step 4: Analyze the graph and determine the point(s) of intersection, if any. The point where the two lines intersect is the solution to the system of equations.
Based on the graph, we can see that the two lines intersect at the point (4, 2). Therefore, the solution to the system of equations is X = 4 and y = 2.
It is worth noting that the system of equations is consistent, meaning it has a single solution.