5D Solving Equations with the Multiplication Property of Equality

Look at the equation 4c = 17. You probably cannot guess what number we can plug in for the variable that will make the equation true. Also subtracting 4 will not help. If we subtract 4 we will get 4c – 4 = 17-4. That equation is more complicated, not less. This equation requires the multiplication property in order to solve it.

The multiplication property of equality says that we can multiply or divide both sides of the equation by any non-zero number. Remember, the key to equation solving is doing the opposite of what is being done to your variable. In 4c =17, the variable is being multiplied by 4, so we should do the opposite. The opposite of multiplying by 4 is dividing by 4. Dividing both sides by 4 gives us the following.

$$\begin{array}{l}4c=17\\ \frac{4c}{4}=\frac{17}{4}\end{array}$$

4 divided by 4 is 1 and 17 divided by 4 is 4.25 so we get the following:

$$\begin{array}{l}4c=17\\ \frac{{}^{1}4c}{4}=\frac{17}{4}\\ c=4.25\end{array}$$

Hence the answer is 4.25. Again we can check it by plugging in 4.25 into the original equation and seeing if it is equal. 4(4.25) = 17

Let’s look at another example. $\frac{y}{7}=2.5$

Since we are dividing our variable by 7, we should multiply both sides by 7 in order to isolate the variable.

$$\begin{array}{l}7\left(\frac{y}{7}\right)=7\left(2.5\right)\\ {}^{1}7\left(\frac{y}{7}\right)=7\left(2.5\right)\\ 1y=17.5\end{array}$$

Since 7/7 = 1 and 7x2.5 = 17.5 and 1y = y, we are left with y = 17.5

Let’s look at a third example. $-\frac{2}{3}w=\frac{7}{8}$

The variable is being multiplied by -2/3 so we need to divide both sides by -2/3. If you remember from fractions, dividing by a fraction is the same as multiplying by the reciprocal. So dividing by -2/3 is the same as multiplying by -3/2. So we are going to multiply both sides by -3/2.

$$-\frac{3}{2}(-\frac{2}{3}w)=-\frac{3}{2}\left(\frac{7}{8}\right)$$

Notice the reciprocals multiply to positive 1. So we are left with an answer of $-\frac{21}{16}$ or $-1⁤\frac{5}{16}$.

$$\begin{array}{l}-\frac{3}{2}(-\frac{2}{3}w)=-\frac{3}{2}\left(\frac{7}{8}\right)\\ +\frac{6}{6}w=-\frac{21}{16}\\ 1w=-\frac{21}{16}\\ w=-1⁤\frac{5}{16}\end{array}$$

Solve the following equations with your instructor by using the multiplication property.

Practice Problems Section 5D

Solve the following equations. Simplify all fractions completely.

1. $$2x=22$$

   Check answer

   $$x=11$$

2. $$28=-4y$$

3. $$12m=-72$$

   Check answer

   $$m=-6$$

4. $$-8=40n$$

5. $$-9w=-144$$

   Check answer

   $$w=16$$

6. $$-65=-5y$$

7. $$6m=90$$

   Check answer

   $$m=15$$

8. $$-98=-7a$$

9. $$14v=-70$$

   Check answer

   $$v=-5$$

10. $$13=-39n$$

11. $$-120d=-20$$

    Check answer

    $$d=\frac{1}{6}$$

12. $$180=-15x$$

13. $$34=-2h$$

    Check answer

    $$h=-17$$

14. $$12g=240$$

15. $$-51=-3L$$

    Check answer

    $$L=17$$

16. $$14f=-70$$

17. $$5=20x$$

    Check answer

    $$x=\frac{1}{4}$$

18. $$36c=-4$$

19. $$\frac{2}{7}y=\frac{3}{5}$$

    Check answer

    $$y=2⁤\frac{1}{10}$$

20. $$\frac{4}{5}=-\frac{12}{25}u$$

21. $$-\frac{1}{9}u=-\frac{5}{18}$$

    Check answer

    $$u=2⁤\frac{1}{2}$$

22. $$\frac{4}{9}=-7y$$

23. $$3b=-\frac{1}{5}$$

    Check answer

    $$b=-\frac{1}{15}$$

24. $$-\frac{4}{11}=\frac{16}{33}L$$

25. $$-\frac{3}{8}T=\frac{9}{8}$$

    Check answer

    $$T=-3$$

26. $$-\frac{3}{14}=-\frac{3}{7}h$$

27. $$\frac{6}{13}p=-\frac{14}{13}$$

    Check answer

    $$-2⁤\frac{1}{3}$$

28. $$0.4m=5.2$$

29. $$0.3=-0.24a$$

    Check answer

    $$a=-1.25\text{ or }-1⁤\frac{1}{4}$$

30. $$1.2w=-0.144$$

31. $$1.8=-0.09p$$

    Check answer

    $$p=-20$$

32. $$-0.1d=0.037$$

33. $$-0.47=-2.35x$$

    Check answer

    $$x=0.2\text{ or }\frac{1}{5}$$

34. $$6.156=-1.8n$$

35. $$0.004c=-0.2$$

    Check answer

    $$c=-50$$

36. $$0.2312=6.8b$$

37. $$0.035g=-0.056$$

    Check answer

    $$g=-1.6\text{ or}-1⁤\frac{3}{5}$$

38. $$-12=0.05f$$

39. $$-0.33m=-8.58$$

    Check answer

    $$m=26$$

40. $$35=-5n$$

41. $$15h=75$$

    Check answer

    $$h=5$$

42. $$\frac{7}{8}=\frac{2}{3}x$$

43. $$-\frac{1}{5}p=-\frac{7}{15}$$

    Check answer

    $$p=\frac{7}{3}$$

44. $$\frac{8}{15}=\frac{7}{12}y$$

45. $$2.4m=-0.288$$

    Check answer

    $$m=-0.12$$

46. $$\frac{b}{5}=-45$$

47. $$\frac{n}{7}=14$$

    Check answer

    $$n=98$$

48. $$0.0005=0.0003x$$

49. $$-\frac{m}{12}=4$$

    Check answer

    $$m=-48$$

50. $$9.2x=-64.4$$

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