1. Tess McMahon

2. What is a Linear System? A linear system is two or more linear equations that use the same variables Ex: X + 2y = 5 Equation 1 2x – 3y = 3 Equation 2 The solution of the linear system is a pair of numbers (a = x and b = y) that make BOTH equations true

3. How to Solve Linear Equations Graphing Substitution Linear Combinations

4. Graphing a Linear System Graph both equations on the same graph Equation 1 Equation 2 x + 2y = 5 2x – 3y = 3 The point where the graphs cross each other is the solution of the equation POINT OF INTERSECTION: (3, 1)

5. Solving Linear Systems by Substitution

6. Step-by-Step The Equations: – Equation 1: x + 2y = 5 – Equation 2: 2x – 3y = 3

7. Step 1 1. Put x and y on opposite sides of the equal sign in Equation 1 x + 2y = 5 -2y -2y x = 5 – 2y

8. Step 2 2. Substitute the x in Equation 2 with the answer you found in step 1. You want to get rid of the x’s to solve for y: 2x – 3y = 3 2(5 – 2y) – 3y = 3 10 - 4y - 3y = 3 10 – 7y = 3 - 10 - 10 -7y = -7 -7 Y = 1

9. Step 3 3. Plug in the value of y from Equation 2 into Equation 1 to solve for x: x + 2y = 5 x + 2 (1) = 5 x + 2 = 5 - 2 -2 x = 3

10. The Solution: (3,1)(3,1)

11. Linear Combinations In linear combinations, we must: 1. Multiply both equations by a constant (if necessary) 2. Add the resulting equations Basically, the goal is to use addition to cancel out a variable (y or x) to solve the linear system

12. Steps in Linear Combinations Without Having to Multiply 1. Write both equations: Equation 1: 4x +3y = 16 Equation 2: 2x – 3y = 8

13. Steps in Linear Combinations Without Having to Multiply 2. Add the equations together to get rid of y. Because the constant before y in Equation one is 3 and the constant before y in Equation two is -3, when you add them together, y is cancelled out. 4x + 3y = 16 + 2x – 3y = 8 6x = 24 x = 4

14. Steps in Linear Combinations Without Having to Multiply 3. Plug in 4 for the value of x in Equation 1 (you could also use Equation 2). Now you are trying to find the value of y. 4x + 3y = 16 4(4) + 3y = 16 16 + 3y = 16 -16 -16 3y = 0 y = 0

15. Solution (4, 0)

16. Steps in Linear Combinations that Involve Multiplication 1. Write both equations: Equation 1: 6x + 2y = 2 Equation 2: -3x + 3y = -9 2. If you look at the two, you’ll notice that if you multiply Equation 2 by 2, the constant before x will equal -6.

17. Steps in Linear Combinations that Involve Multiplication 3. Multiply Equation 2 by 2 so that the x-intercept will cancel out: 6x + 2y = 2 2(-3x + 3y) = (-9)2 Equation 1: 6x + 2y = 2 Equation 2 (Modified): -6x + 6y = -18

18. Steps in Linear Combinations that Involve Multiplication 4. Now add the new Equation 2 and the old Equation 1 to get rid of x: 6x + 2y = 2 +(-6x + 6y = -18) 8y = - 16 8 8 y = -2

19. Steps in Linear Combinations that Involve Multiplication 5. Plug -2 in for y in Equation 1. Now you are trying to find the value for x. 6x + 2y = 2 6x + 2(-2) = 2 6x - 4 = 2 + 4 + 4 6x = 6 x = 1

20. Solution (1, -2)

21. How Can We Use Linear Systems? One of the reasons we use linear systems is to solve word problems.

22. Solving Word Problems Using Linear Combinations Example: You combine 2 solutions to form a mixture that is 40% acid. One solution is 20% acid (solution A) and the other is 50% acid (solution B). If you have 90 ml of the mixture, how much of each solution was mixed?

23. Solving Word Problems Using Linear Combinations Before you freak out, figure out what they are looking for: ‘how much’ = VOLUME. Therefore, x and y must be the volumes of solution A and solution B. Volume of Solution A  x Volume of Solution B  y Volume of Solution A and Solution B = 90 ml So, you’re first equation is: x + y = 90

24. Solving Word Problems Using Linear Combinations The other equation will be about the other part of the problem: acidity. Acid in Solution A: 20% of x (in ml) Acid in Solution B: 50% of y (in ml) Acid in Mixture: 40% of 90 ml 0.40 x 90ml = 36 ml Therefore Equation 2 will be: 0.20x +.50y = 36

25. Solving Word Problems Using Linear Combinations Now we must solve the equations using the same steps we used before: Equation 1: x + y = 90 Equation 2: 0.2x + 0.50y = 36 Can YOU solve it?

26. Linear Systems with Infinitely Many Solutions or None at All Some Linear Systems are special and have either no solution OR an infinite number of solutions:

27. Linear Systems with Infinitely Many Solutions or None at All Parallel Lines have NO solution because their lines NEVER intersect.

28. Linear Systems with Infinitely Many Solutions or None at All Some linear systems have equations that some out to equal the same line. Because it is the same line, the two equations are intersecting at every point on their graphs.

29. Linear Systems with Infinitely Many Solutions or None at All You can prove that a system has infinitely many solutions or no solution using the same methods to solve the systems as we used before: Graphing Substituting Linear Combinations

30. Graphing Systems of Linear Inequalities A system of linear inequalities is two or more linear inequalities that use the same variables (i.e. they both use x and y).

31. Graphing Systems of Linear Inequalities A graph of a linear inequality in two variables is a half-plane. The boundary line of the half-plane is dashed if the inequality is and solid if the inequality is ≤ or ≥.

32. Graphing Systems of Linear Inequalities Example: x + y < 3 x + 4y ≥ 0

33. Graphing Systems of Linear Inequalities To graph the system simply put both graphs on the same paper. The shaded regions will overlap: