Consider the system of equations x + y = 8 and 2x – 2y = 4. Find the point of intersection for this system of equations and explain what the coordinates of that point represent. Graph at least three points for each equations then label the point of intersection.
To find the point of intersection, we can solve the system of equations using either the substitution or elimination method.
Using the substitution method:
1. Solve one equation for one variable in terms of the other variable.
From the first equation, we have x = 8 - y.
2. Substitute this expression for x in the second equation.
2(8 - y) - 2y = 4
16 - 2y - 2y = 4
16 - 4y = 4
-4y = -12
y = 3
3. Substitute this value for y back into one of the original equations to solve for x.
x + 3 = 8
x = 5
Therefore, the point of intersection is (5, 3).
To graph the equations, we can create tables of values for x and y and plot the corresponding points.
For the first equation, x + y = 8:
x | y
---------
0 | 8
1 | 7
2 | 6
For the second equation, 2x - 2y = 4:
x | y
---------
-2 | 3
0 | 2
2 | 1
Plotting these points on a coordinate plane, we can see that the point of intersection is (5, 3).
The coordinates of the point of intersection represent the values of x and y that satisfy both equations simultaneously. In other words, it is the solution to the system of equations.