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Two-Step Equations and Properties of Equality

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Solving Two Step Equations

The Cell-U-Lar cell phone company has a data plan where 2.5 GB are included for $79.00 per month. For every 10 MB (there are 1024 MB in 1 GB) over that amount, the company charges an additional $3.00. Last month, Taylor's cell phone bill was $115. How many megabites of data was he over the limit?

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James Sousa: Solving Two Step Equations

Guidance

Now, we will compound steps to solve slightly more complicated equations that the previous concept. We will continue to undo the operations that are in an equation. Remember that order matters. To undo the operations, work in the opposite order of the Order of Operations.

Example A

Solve 4x-9=11

Solution: Now we need to undo two operations. Therefore, we need to get everything on the other side of the equals sign to solve for x . First, add 9 to both sides.

& 4x - \bcancel{9} = 11\\& \underline{\quad \ + \bcancel{9} \ \ +9}\\& \quad \ \ 4x=20

Now, undo the multiplication. Divide both sides by 4 to solve for x .

\frac{\cancel{4}x}{\cancel{4}} &= \frac{20}{4}\\x &= 5

Check your answer: 4(5) - 9 = 20 - 9 = 11

Example B

Solve -\frac{1}{2}g+5=14 .

Solution: First, subtract 5 from both sides.

& -\frac{1}{2}g+\bcancel{5}=14\\& \underline{\qquad \ \ -\bcancel{5} \ \ -5 \; \;}\\& \quad \ -\frac{1}{2}g=9

Multiplying by one-half is the same as dividing by 2. To undo this fraction, multiply both sides by –2.

 -\frac{1}{2}g &= 9\\-\bcancel{2} \cdot -\frac{1}{\bcancel{2}}g &= 9 \cdot -2\\g &= -18

Check your answer: -\frac{1}{2}(-18)+5=9+5=14

Example C

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

Solution: Even though this problem involves fractions, it is still solved the same way. First, add \frac{2}{3} to both sides and then multiply both sides by the reciprocal of \frac{3}{4} .

& \frac{3}{4}x-\bcancel{\frac{2}{3}}=\frac{7}{3}\\& \underline{\quad \ + \bcancel{\frac{2}{3}} \ +\frac{2}{3} \; \; \; \; \; \; \;}\\& \quad \ \ \frac{3}{4}x=\frac{9}{3}=3\\& \ \ \xcancel{\frac{4}{3} \cdot \frac{3}{4}} x=\bcancel{3} \cdot \frac{4}{\bcancel{3}}\\& \qquad \ \ x=4

Check your answer: \frac{3}{4}(4)-\frac{2}{3}=3-\frac{2}{3}=\frac{9}{3}-\frac{2}{3}=\frac{7}{3} .

For two-step equations:

  1. First, undo any addition or subtraction.
  2. Second, undo any multiplication or division.

Intro Problem Revisit Writing an equation for the phone bill, we have: C=79+3m where C is the cost, and m is the number of 10 megabites over the 2.5 GB. So, when we get our final answer, we need to multiply it by 10 to get the total number of megabites over the included 2.5 GB amount. Set C = $115 and solve for m .

 \bcancel{79} + 3m &= 115\\ \bcancel{-79} + 3m &= -79 \\3m &=36\\\frac{\cancel{3}m}{\cancel{3}} &= \frac{36}{3}\\m &= 12\\12 \cdot 10 = 120

Taylor is 120 MB over his data limit for the month.

Guided Practice

Solve the following equations and check your answer.

1. \frac{x}{-6}-7=-11

2. 18=6-\frac{2}{5}y

Answers

1. Follow the steps from above. First add 7 to both sides and then multiply both sides by –6.

& \ \frac{x}{-6} -\bcancel{7} = -11\\& \underline{\qquad +\bcancel{7} \quad \ +7 \; \; \; \; \; \; \; \;}\\& \qquad \frac{x}{-6}=-4\\& -\bcancel{6} \cdot \frac{x}{-\bcancel{6}}=-4 \cdot -6\\& \qquad \quad x=24

Check your answer: \frac{24}{-6}-7=-4-7=-11

2. Again, follow the steps. Here, we will subtract 6 and then multiply by -\frac{5}{2} .

& \qquad \quad 18=\bcancel{6}-\frac{2}{5}y\\& \qquad \ \ \underline{-6 \ -\bcancel{6} \; \; \; \; \; \; \; \; \; \; \;}\\& \qquad \ \ 12=-\frac{2}{5}y\\& -\frac{5}{2} \cdot 12=-\frac{\bcancel{2}}{\cancel{5}}y \cdot - \frac{\cancel{5}}{\bcancel{2}}\\& \quad \ -30=y

Check your answer: 6-\frac{2}{5}(-30)=6+12=18

Practice

Solve the following equations and check your answer. Reduce any fractions.

  1. 2x-5=-17
  2. -4x+3=-5
  3. -1=\frac{x}{2}-6
  4. \frac{1}{3}x+11=-2
  5. \frac{3}{4}=\frac{1}{2}-\frac{1}{8}x
  6. -18=\frac{x}{-5}-3
  7. \frac{5}{6}x+4=29
  8. -11x+4=125
  9. 6-x=-22
  10. \frac{2}{7}x+8=20
  11. \frac{4}{5}=-\frac{2}{5}+\frac{3}{2}x
  12. 15-\frac{x}{-9}=21
  13. 1.4x-5.6=2.8
  14. 14.4=-2.7x - 1.8
  15. -\frac{5}{4}=\frac{1}{6}x+\frac{3}{12}

When dealing with fractions, one way to “get rid” of them is to multiply everything in the equation by the Least Common Denominator (LCD) for the entire equation. Try this technique with the following equations. We will get you started in #16.

  1. \frac{3}{8}x-\frac{2}{5}=\frac{7}{4} The LCD of 8, 5, and 4 is 40. 40 \left(\frac{3}{8}x-\frac{2}{5}=\frac{7}{4}\right) . Distribute the 40 and solve.
  2. \frac{10}{3}x+\frac{3}{4}=\frac{5}{2}
  3. \frac{9}{10}x-\frac{1}{2}=\frac{2}{3}
  4. Solve #15 using the LCD method.
  5. Which method is easier for you to solve equations with fractions?

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