Module 1 ~ Topic 1 Solving Equations

1. Module 1 ~ Topic 1 Solving Equations • Table of Contents • Slides 6-14: Solving Linear Equations • Slides 15-29 : Practice Questions • Audio/Video and Interactive Sites • Slides 2: Algebra Cheat Sheet • Slides 3: Graphing Calculator Use Guide • Slide 5: Video • Slide 7: Gizmos

2. Algebra Cheat Sheet You may want to download and print this sheet for reference throughout the course.

3. Special Instructions • This module includes graphing calculator work. • Refer to the website, TI-83/84 calculator instructions, for resources and instructions on how to use your calculator to obtain the results we do throughout the lessons. I suggest that you bookmark this site if you haven’t done so already. • Take careful notes and following along with every example in each lesson. I encourage you to ask me questions and think deeply as you’re studying these concepts!

4. Topic #1: Solving Linear Equations • Many real-life phenomena can be described by linear functions. It is important to learn how to solve equations involving such functions to provide us with information about these phenomena. Definition: A linear equation in one variable is an equation that can be written in the form ax + b = c, where a, b and c are real numbers, and a ≠ 0. (The letter x is often used as the variable, but it is not required to be the variable.)

5. Video Break!!!!! Click on this link to watch videos on solving equations.

6. Recall General Rules Order of Operations: PEMDAS Multiplication/Division is done in order, left to right Addition/Subtraction is done in order, left to right Solving Equations First, you must know what you are solving for so you can isolate it. To do that: Take care of any exponents/FOIL or distribution/simplification Get common denominators if necessary Combine like terms on each side of the equal sign Addition/Subtraction across = is done next to isolate the variable Multiplication/Division across =is done last and the variable should now be isolated

7. Gizmos Gizmo: Modeling 1-Step Equations A Gizmo: Modeling 1-Step Equations B Gizmo: Modeling 2-Step Equations Gizmo: Solving 2-Step Equations Gizmo: Solving Formulas

8. How to Solve Equations As with any journey, you need to know where you want to end up before you start off, or you will never get there! Ex: Solve • Figure out what you are solving for. • In this case we want to end up with x = # 2) Move everything that is NOT x to the other side. When solving equations, do all addition/subtraction first. Then all multiplication/ division last. Make sure you are where you wanted to end up, x = some number.

9. Example 1 Solving an equation in one variable algebraically means to find the value of the variable that makes the mathematical statement trueusing appropriate algebraic operations. Example 1: Solve the equation 5x - 4 = 0 . 5x – 4 = 0 + 4 +4 5x = 4 5x = 4 5 5 x = This means makes 5x – 4 = 0 true.

10. One of the great things about solving equations is that we can always check our solutions! We do this by plugging the value(s) we get for our variable into the original equation and verify that we have a true statement. Example 1: Solve the equation 5x - 4 = 0 . Check: 5x – 4 = 0 + 4 +4 5x = 4 5x = 4 5 5 x = We simplify and get 4-4, which is 0. This is what our original equation stated. This is true. Original Equation Now, plug in the value of x that you just found

11. Example 2 Example 1: Solve the equation 5(h – 2) = -4(3-h). Check: Solution: We have a couple of choices as to how to solve this equation. You may notice that it is not in the form introduced earlier, but it still a linear equation in one variable, since it can be written in the form described above. Our goal is to isolate the variable h, so that it appears on one side of the equation, and its value appears on the other. We will need to distribute on both sides of the equation before we can do that. 5(h - 2) = -4(3 – h) 5((-2) – 2) = -4(3 – (-2)) 5( - 4) = -4 ( 3 + 2) -20 = - 4 (5) -20 = -20 5(h - 2) = -4(3 – h) 5h – 10 = -12 + 4h +10 +10 5h = -2 + 4h 5h = -2 + 4h -4h - 4h h = -2 Since both sides of the equation yield the same result, we know that our answer is correct!

12. Example 1 Example 1: Solve the equation Remember: Solving Equations First, you must know what you are solving for so you can isolate it. To do that: Take care of any exponents/FOIL or distribution/simplification Get common denominators if necessary Combine like terms on each side of the equal sign Addition/Subtraction across = is done next to isolate the variable Multiplication/Division across =is done last and the variable should now be isolated

13. Example 1: Solution -45 -45 -10m -10m 140 140

14. Check: We find that both sides of the equation give us the same result when we plug our answer in, which means that we obtained the correct result!

15. Practice Examples Example 2: Solve the equation 3p + 2 = 0 . Example 3: Solve the equation -7m – 1 = 0. Example 4: Solve the equation 14z – 28 = 0. Solutions on next slide. Solve these on your own first.

16. Practice Examples Answers Example 2: Solve the equation 3p + 2 = 0 . Check: Example 3: Solve the equation -7m – 1 = 0. Check: Example 4: Solve the equation 14z – 28 = 0. Check:

17. More Practice Examples Example 5: Solve the equation . Example 6: Solve the equation . Solve these on your own first. Solutions on next slide.

18. More Practice Examples - Answers Example 5: Solve the equation . Example 6: Solve the equation .

19. More Practice Examples Example 7: Solve the equation . Example 8: Solve the equation . Solve these on your own first. Solutions on next slide.

20. More Practice Examples - Answers Example 7: Solve the equation . Example 8: Solve the equation .

21. More Practice Examples Example 9: Solve the equation for m. . Example 10: Solve the equation for . Solve these on your own first. Solutions on next slide.

22. More Practice Examples Example 9: Solve the equation for m. . Example 10: Solve the equation for .

23. More Practice Examples Example 11: Solve the equation for x. No Solution Obviously, , so the answer is No Solution

24. Word Problem Examples Example: The relationship between ºC and ºF can be represented by the equation where F is the number of degrees Fahrenheit, and C is the number of degrees Celsius. • Solve the above equation for C. • Convert 98oF into degrees Celsius

25. Solution a): We want to isolate C on one side of the equation. So, we apply the following operations on our original equation. -32 -32 or Now we have an equation that allows us to compute degrees Celsius if we knew degrees Fahrenheit.

26. Solution b): We plug 98 in for F and solve for C. -32 -32 9 9 So, 98ºF is approximately 36.67ºC.

27. Example: When you buy a new car, they say that the value of the car depreciates as soon as you drive it off the lot! Accountants use the following equation to measure depreciation of assets: where … D is the depreciation of the asset per year, C is the initial cost of the asset, S is the salvage value, and L is the asset’s estimated life. a. What is the salvage value of a machine that cost a company $40,000 initially, has an annual depreciation of $3000, and an estimated life of 10 years? b. Solve the original equation for S, the salvage value, in general.

28. Solution a): We plug 30,000 in for C, 2000 for D, and 10 for L. We then solve for S. So, the salvage value for the machine is $10,000.

29. Solution b): We want to isolate the variable S, treating all of the other letters in the equation as constants. This equation allows us to calculate the salvage value for any asset, given the initial cost, estimated life, and depreciation value. -C -C