solve each system graphically. if a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this.

Question

solve each system graphically. if a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this.

y = 2 - x
2x + 2y = 4

Answer 1

y = 2-x or x+y = 2

2x+2y = 4
x+y = 2

the two equations are the same
so there is an infinite number of solutions.

Answer 2

To solve the system of equations graphically, we need to find the points where the two equations intersect on a graph.

First, let's graph the first equation: y = 2 - x.

To do this, we can start by assigning some values to x and calculate the corresponding values of y. Then, we can plot these points on a graph and connect them to get a straight line.

Let's choose three values for x: -2, 0, and 2.

For x = -2, y = 2 - (-2) = 4. So we have the point (-2, 4).
For x = 0, y = 2 - 0 = 2. So we have the point (0, 2).
For x = 2, y = 2 - 2 = 0. So we have the point (2, 0).

Plotting these three points on a graph and connecting them will give us a straight line.

Next, let's graph the second equation: 2x + 2y = 4.

To graph this equation, we can rearrange it to the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

First, let's solve the equation for y:

2x + 2y = 4
2y = -2x + 4
y = -x + 2

So the equation can be rewritten as y = -x + 2.

Using the same method as before, let's assign some values to x and calculate the corresponding values of y. Then, we can plot these points on the graph.

Choose three values for x: -2, 0, and 2.

For x = -2, y = -(-2) + 2 = 4. So we have the point (-2, 4).
For x = 0, y = -0 + 2 = 2. So we have the point (0, 2).
For x = 2, y = -2 + 2 = 0. So we have the point (2, 0).

Plotting these three points on the graph and connecting them will give us a straight line.

Now, we have two lines on the graph. The solution of the system of equations is the point where these two lines intersect.

By looking at the graph, we can see that the two lines intersect at the point (1, 1).

Therefore, the solution to the system of equations is x = 1 and y = 1.

If a system of equations has an infinite number of solutions, it means the two equations represent the same line. In this case, the solution set can be written in set-builder notation as:

{(x, y) | x - y = 2}

If a system of equations has no solution, it means the two equations represent parallel lines that never intersect. In this case, we would state that the system has no solution.