1. 11.6 Systems of Equations

2. 11.6 Systems Vocabulary
System of Equations: a set of two or more equations that contain two or more variables.
Solution of System of Equations: set of values that are solutions of all of the equations.
If the system has two variables, the solutions can be written as ordered pairs, like (x,y).

3. 11.6 Systems Background
Systems of Equations have at least 2 equations and their graphs may intersect. These intersection points are the solutions to the system. If they intersect at one point there is one solution, and crossing at 2 points means two solutions. If they are parallel they never intersect so there are no solutions. If they are both on the same line then they share all of their points, so there are infinite solutions.

4. 11.6 Systems Example 1a Solve the system of two equations.
y = x + 3, and y = 2x + 5
Since both equations equal y, they must be equal to each other. So,
x + 3 = 2x + 5, subtract x from each side
- x x
3 = x + 5, subtract 5 from each side
- 2 = x, use this info to find y (next slide)

5. 11.6 Systems Example 1b
Use x = -2 to find y. Substitute -2 for x is one of the original equations. Try y = x + 3.
Y = (-2) + 3
Y = 1
Since x = -2 and y = 1, our solution is the point:
(-2,1)

6. 11.6 Systems Example 2 Try y = 3x + 8, and y = -7 + 3x. So,
3x + 8 = x, subtract 3x from both sides
- 3x x
8 = -7, since 8 does NOT equal -7, this statement is false. So,
This system of equations has no solution.

7. 11.6 Systems Example 3a
Solve 3x + y = 8, and 6x + 2y = 16, simplify so that both equations start with y = something:
3x + y = 8, subtract 3x from each side of Eq.1
3x (– 3x) + y = 8 (- 3x), so
y = 8 – 3x, now simplify Equation 2
6x + 2y = 16, subtract 6x from each side
6x (- 6x) + 2y = 16 (- 6x), so 2y = 16 – 6x, divide
2y/2 = 16/2 – 6x/2, so y = 8 – 3x

8. 11.6 Systems Example 3b Now put the right sides of answers together.
8 – 3x = 8 – 3x, add 3x to each side
8 – 3x (+ 3x) = 8 – 3x (+ 3x), so
8 = 8
Since this statement is always true, then the system of equations has an infinite number of solutions.