To solve a system of equations, you need to find the values of the variables that satisfy both equations simultaneously. There are different methods to solve systems of equations, including substitution, elimination, and graphing. Here's an example of using substitution:
Solve the system of equations:
2x + y = 5
x - y = 1
From the second equation, we get x = y + 1. Substitute this expression for x into the first equation:
2(y + 1) + y = 5
Simplifying and solving for y, we get:
3y + 2 = 5
3y = 3
y = 1
Then, substitute y = 1 into x = y + 1 to get:
x = 2
So the solution to the system of equations is x = 2 and y = 1.
To solve a system of inequalities, you need to find the values of the variables that satisfy both inequalities simultaneously. There are different methods to solve systems of inequalities, including graphing and substitution. Here's an example of using graphing:
Solve the system of inequalities:
x + y ≤ 3
x - y > 1
First, graph the boundary lines of each inequality. For the first inequality, the boundary line is x + y = 3, which is a line with intercepts (3, 0) and (0, 3). To decide which side of the line to shade, test a point that is not on the line, such as (0, 0):
0 + 0 ≤ 3, which is true
Therefore, shade the side of the line that contains the origin.
For the second inequality, the boundary line is x - y = 1, which is a line with intercepts (1, 0) and (0, -1). To decide which side of the line to shade, test a point that is not on the line, such as (0, 0):
0 - 0 > 1, which is false
Therefore, shade the side of the line that does not contain the origin.
The shaded region that satisfies both inequalities is the region that is below the line x + y = 3 and to the right of the line x - y = 1. This region is a triangle with vertices (1, 2), (2, 1), and (2, 2).
I hope this explanation helps you with your homework!