1. How does a the graph of two coinciding lines look?

2. How many solutions do two graphs with the same slope and the same y-intercept have?

3. A system of equations that has exactly one solution. x + y = 7
x − y = 1
Solution:(4, 3)

4. A method used to solve systems of equations by solving an equation for one variable and substituting the resulting expression into the other equation(s).

y = 2x
y = x + 5

Step 1
y = 2x
y = x + 5

Step 2
plug 2x into y = x + 5 for y
2x = x + 5

Step 3
solve for x
2x − x = x − x + 5
x = 5

Step 4
plug x = 5 into y = 2x
y = 2(5)
y = 10

Step 5
solution
(5, 10)

USE WHEN...
A variable in either equation has a coefficient of 1 or −1.
Both equations are solved for the same variable.
Either equation is solved for a variable.

EXAMPLE
x + 2y = 7
x = 10 − 5y

or
x = 2y + 10
x = 3y + 5

5. A set of two or more linear equations containing two or more variables.

2x + 3y = −1
x − 3y = 4

6. For y = (2/3)x + 6 and y = (2/3) + 6; how many points do these equations have in common?

7. A method used to solve systems of equations in which one variable is eliminated by adding or subtracting two equations of the system.

x − 2y = −19
5x + 2y = 1

Step 1
x − 2y =−19
+5x + 2y = 1
∴ 6x + 0 =>−18

Step 2
6x = −18
6x ÷ 6 = −18 ÷ 6
x = −3

Step 3
plug x = -3 into x − 2y = −19
now, solve −3 − 2y = −19 for y
−3+ 3 − 2y = −19 + 3
−2y = −16
y = 8

Step 4
solution
(−3,8)

USE WHEN...
Both equations have the same variable with the same or opposite coefficients.
A variable term in one equation is a multiple of the corresponding variable term in the other equation.

EXAMPLE
3x + 2y = 8
5x + 2y = 12

or
6x + 5y = 10
3x + 2y = 15

8. An ordered pair that satisfies each equation in the system.

9. A method of solving a system of equations when you solve one equation for a variable, substitute that expression into the other equation and solve, and then use the value of that variable to find the value of the other variable.

10. A system of equations that has infinitely many solutions.

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

11. How does the graph of two parallel lines look?

12. Any ordered pair that satisfies all the equations in the system.

x + y = −1
−x + y = −3
Solution: (1, −2)

13. A set of two or more linear inequalities containing two or more variables.

2x + 3y > −1
x − 3y ≤ 4

14. 1) graphing
2) substitution
3) elimination

15. How many solutions do two lines with different slopes have?

16. How many solutions do two graphs with the same slope and different y-intercepts have?

17. An ordered pair or ordered pairs that make the inequality true.
Inequality: 3x + 2y ≥ 6

18. For y = 3x - 5 and y = -3x + 8; how many points do these equations have in common?

19. How does a the graph of two intersecting lines look?

20. A method of solving systems of equations where you put both equations in standard form, add or subtract the equations to eliminate a variable and solve for the variable left, and then use that number to find the value of the other variable.

21. For y = (1/2)x + 6 and y = (1/2)x - 3.3; how many points do these equations have in common?