5C Solving Equations with the Addition Property

Solving equations is a useful tool for determining quantities. In this section we are going to explore the process and properties involved in solving equations.

For example, in business we look at the break-even point. This is the number of items that need to be sold in order for the company’s revenue to equal the cost. This is the number of items that must be sold so that the company is neither losing money nor earning a profit. For example a company that makes blue-ray DVD players has costs equal to 40x+12000 where x is the number of blue-ray DVD players made. The equipment needed to make the DVD players was $12000 and it costs about $40 for the company to make 1 DVD player. They sell the DVD players for $70 so their revenue is 70x where x is the number of players sold. The break-even point will be where costs = revenue (40x + 12000 = 70x). How many DVD players do they need to sell to break even?

To solve problems like this we need to learn how to solve equations like 40x + 12000 = 70x. To solve an equation, we are looking for the number or numbers we can plug in for the variable that will make the equation true. For example, try plugging in some numbers for x in the break-even equation and see if it is true. If we plug in 100, we get the following: $40(100)+12000=70(100)$. But that is not true! $4000+12000\ne 7000$ so 100 is not the solution. If we plug in 400 we get the following: $40(400)+12000=70(400)$. This is true since $16000+12000=28000$. The two sides are equal! So the company needs to make and sell 400 blue-ray DVD players to break even. After 400, they will start to turn a profit.

As you can see sometimes we can guess the answer to an equation. If you cannot guess the answer, then we need to have ways of figuring out the answer.

Addition Property of Equality

How do we find the answer to an equation when we cannot guess the answer? One property that is very helpful is the addition property. The addition property says that we can add or subtract the same number or term to both sides of an equation and the equation will remain true.

For example look at the equation $w+19=-4$. You may or may not be able to guess the number we can plug in for w that makes the equation true. The key is to add or subtract something from both sides so that we isolate the variable w. In equation solving, it is all about opposites. Do the opposite of what is being done to your letter. Since we are adding 19 to our variable w, let’s subtract 19. Look what happens if we subtract 19 from both sides and simplify.

$$\begin{array}{l}w+19-19=-4-19\\ w+0=-4+-19\\ w=-23\end{array}$$

First notice, that we had to subtract the same number from both sides. If you only subtracted 19 from the left side of the equation, the equation would no longer be true. Also subtracting 19 is the same as adding -19 to both sides. This helps when dealing with negative numbers. This shows us that the number we can plug in for w that makes the equation true is -23. How can we check if that is the correct answer? We plug in -23 for w in the original equation and see if the two sides are equal. $-23+19=-4$ is true so -23 is the correct answer! Notice also that this is a conditional equation and is only true when w = -23 and false for any other number.

Try to solve the following equations with your instructor using the addition property. Be sure to check your answers.

Practice Problems Section 5C

Solve the following equations with the addition property.

1. $$x+9=23$$

   Check answer

   $$x=14$$

2. $$m-12=-5$$

3. $$y-6=-17$$

   Check answer

   $$y=-11$$

4. $$13=v+-7$$

5. $$-18=w-15$$

   Check answer

   $$w=-3$$

6. $$h-13=-24$$

7. $$m+0.13=0.58$$

   Check answer

   $$m=0.45$$

8. $$n-\frac{1}{2}=\frac{3}{8}$$

9. $$-1.19=-2.41+T$$

   Check answer

   $$T=1.22$$

10. $$\frac{4}{5}=\frac{2}{7}+c$$

11. $$x-74=-135$$

    Check answer

    $$x=-61$$

12. $$y+-53=-74$$

13. $$c-8.14=6.135$$

    Check answer

    $$c=14.275$$

14. $$d+\frac{1}{5}=\frac{3}{4}$$

15. $$-630=-440+p$$

    Check answer

    $$p=-190$$

16. $$39=y+86$$

17. $$n+0.0351=-0.0427$$

    Check answer

    $$n=-0.0778$$

18. $$-\frac{1}{10}=\frac{3}{4}+L$$

19. $$a+4.3=-1.2$$

    Check answer

    $$a=-5.5$$

20. $$\frac{4}{5}+m=\frac{1}{2}$$

21. $$\frac{5}{6}+P=\frac{1}{3}$$

    Check answer

    $$P=\frac{-1}{2}$$

22. $$5.07=d-8.1$$

23. $$267=x-243$$

    Check answer

    $$x=510$$

24. $$y+\mathrm{14,320}=\mathrm{145,208}$$

25. $$y+7=-29$$

    Check answer

    $$y=-36$$

26. $$-67=p-78$$

27. $$w-\frac{7}{8}=\frac{5}{12}$$

    Check answer

    $$w=\frac{31}{24}$$

28. $$t-\frac{1}{9}=-\frac{3}{4}$$

29. $$x-\frac{4}{3}=\frac{2}{7}$$

    Check answer

    $$x=\frac{34}{21}$$

30. $$5.013+y=-2.13$$

31. $$2.3=\frac{23}{10}+y$$

    Check answer

    $$y=0$$

32. $$7221+u=\mathrm{18,239}$$

33. $$q-45.2=-27$$

    Check answer

    $$q=18.2$$

34. $$\frac{9}{8}=\frac{1}{3}+w$$

35. $$0.4+x=\frac{1}{2}$$

    Check answer

    $$x=0.1\text{ or }\frac{1}{10}$$

36. $$f-\frac{3}{7}=\frac{9}{4}$$

37. $$m+\frac{8}{7}=-\frac{2}{9}$$

    Check answer

    $$m=\frac{-86}{63}$$

38. $$p+12.8=-4.97$$

39. $$\frac{3}{4}+x=0.75$$

    Check answer

    $$x=0$$

40. $$8.001=m-0.12$$

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.