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Multi-step equations review

To solve an equation we find the value of the variable that makes the equation true. For more complicated, fancier equations, this process can take several steps.
When solving an equation, our goal is to find the value of the variable that makes the equation true.

Example 1: Two-step equation

Solve for x.
3, x, plus, 7, equals, 13
We need to manipulate the equation to get x by itself.
3x+7=133x+77=1373x=63x3=63x=2\begin{aligned} 3x+7&=13 \\\\ 3x+7\redD{-7}&=13\redD{-7} \\\\ 3x&=6 \\\\ \dfrac{3x}{\redD{3}}&=\dfrac{6}{\redD{3}} \\\\ x&=2 \end{aligned}
We call this a two-step equation because it took two steps to solve. The first step was to subtract 7 from both sides, and the second step was to divide both sides by 3. Want an explanation of why we do the same thing to both sides of the equation? Check out this video.
We check the solution by plugging start color #e84d39, 2, end color #e84d39 back into the original equation:
3x+7=1332+7=?136+7=?1313=13       Yes!\begin{aligned} 3x+7&=13 \\\\ 3\cdot \redD 2 + 7 &\stackrel?= 13 \\\\ 6+7 &\stackrel?= 13 \\\\ 13 &= 13 ~~~~~~~\text{Yes!} \end{aligned}

Example 2: Variables on both sides

Solve for a.
5, plus, 14, a, equals, 9, a, minus, 5
We need to manipulate the equation to get a by itself.
5+14a=9a55+14a9a=9a59a5+5a=55+5a5=555a=105a5=105a=2\begin{aligned} 5 + 14a &= 9a - 5 \\\\ 5 + 14a \blueD{- 9a} &= 9a - 5 \blueD{- 9a} \\\\ 5 + 5a &= -5 \\\\ 5 + 5a \blueD{-5} &= -5 \blueD{- 5}\\\\ 5a &= -10\\\\ \dfrac{5a}{\blueD5} &= \dfrac{-10}{\blueD5} \\\\ a &= \blueD{-2} \end{aligned}
The answer:
a, equals, start color #11accd, minus, 2, end color #11accd
Check our work:
5+14a=9a55+14(2)=?9(2)55+(28)=?18523=23       Yes!\begin{aligned} 5 + 14a &= 9a - 5 \\\\ 5 + 14(\blueD{-2}) &\stackrel?= 9(\blueD{-2}) - 5 \\\\ 5 + (-28) &\stackrel?= -18 - 5 \\\\ -23 &= -23 ~~~~~~~\text{Yes!} \end{aligned}
Want to learn more about solving equations with variables on both sides? Check out this video.

Example 3: Distributive property

Solve for e.
7, left parenthesis, 2, e, minus, 1, right parenthesis, minus, 11, equals, 6, plus, 6, e
We need to manipulate the equation to get e by itself.
7(2e1)11=6+6e14e711=6+6e14e18=6+6e14e186e=6+6e6e8e18=68e18+18=6+188e=248e8=248e=3\begin{aligned} 7(2e-1)-11 &= 6+6e \\\\ 14e-7 -11&= 6+6e\\\\ 14e-18 &= 6+6e\\\\ 14e-18\purpleD{-6e} &= 6+6e\purpleD{-6e} \\\\ 8e-18&=6\\\\ 8e-18\purpleD{+18} &=6 \purpleD{+18} \\\\ 8e &=24\\\\ \dfrac{8e}{\purpleD{8}}&= \dfrac{24}{\purpleD{8}}\\\\ e &= \purpleD{3} \end{aligned}
The answer:
e, equals, start color #7854ab, 3, end color #7854ab
Check our work:
7(2e1)11=6+6e7(2(3)1)11=?6+6(3)7(61)11=?6+187(5)11=?243511=?2424=24       Yes!\begin{aligned} 7(2e-1)-11 &= 6+6e \\\\ 7(2(\purpleD{3})-1) -11&\stackrel?= 6+6(\purpleD{3}) \\\\ 7(6-1)-11 &\stackrel?= 6+18 \\\\ 7(5)-11&\stackrel?=24 \\\\ 35-11&\stackrel?=24 \\\\ 24 &=24 ~~~~~~~\text{Yes!} \end{aligned}
Want to learn more about solving equations with the distributive property? Check out this video.

Practice

Problem 1
Solve for b.
4, b, plus, 5, equals, 1, plus, 5, b
b, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Want more practice? Check out these exercises:

Want to join the conversation?

  • blobby green style avatar for user watermelongrin
    how did you get 5/3b?
    (47 votes)
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    • duskpin seed style avatar for user Amos Liberal
      So as we can see the question starts off as:
      2/3b + 5 = 20-b
      first, add +b to both sides of the equation
      (addition property of equality). Next:
      We got 5/3 by adding fractions:
      2/3+1/1.
      Step 1: Multiply the denominators (x/3)
      Step 2: Cross multiply the numerators and denominators (2x1 and 3x1)
      Step 3: Add the two products together (2x1=2, 3x1=3 therefore, add 2+3). WITHOUT touching the denominator!
      Step 4: 5/3b + 5 = 20. Subtract 5 from both sides of the equation to cancel out 5.
      Step 5. divide 5/3 to 15. Keep change Flip
      Keep the fraction change the division sign to multiplication and flip the second fraction (example 2/3 to 3/2). So, 5/3 to 3/5 and multiply both sides of the equation, lastly, your answer is 4.
      (10 votes)
  • blobby green style avatar for user Ayah Khanafer
    What do I do if the variable is equal to 0? How do I check my answer?
    (21 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      To check your answer, just plug 0 in for the variable.

      For example, let's say you solved this equation:
      9x - 1 = 5x - 1
      9x - 5x = 1 - 1
      4x = 0
      x = 0
      It looks like the variable is equal to 0.

      Now, double check your answer with the original equation by replacing all the x's with 0's:
      9x - 1 = 5x - 1
      9(0) - 1 = 5(0) - 1
      0 - 1 = 0 - 1
      -1 = -1
      The equation is true, so 0 is valid.

      Hope this helps!
      (35 votes)
  • winston default style avatar for user Colin
    These comments be getting answers years later. Like bro they already progressed into High School or College, they most likely know after then.
    (20 votes)
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  • starky sapling style avatar for user Anna Russum
    This type of math is a little confusing because sometimes when u have a -3x you divide or subtract at least i think so i don't know why but my teacher said that and can some one explain it please?
    (8 votes)
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  • mr pants pink style avatar for user sreyar27
    how do you solve an equation like 0.5(5-7x) = 8-(4x+6)
    (5 votes)
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    • stelly blue style avatar for user Kim Seidel
      1) Use distributive property to remove the parentheses. Distribute the 0.5 and the "-"
      2) On the right side, you will have 2 terms that are like terms. Combine them.
      3) Move all the "x" terms to the same side of the equation
      4) Move the constants to the opposite side from the x's
      5) Then divide by the coefficient of "x" (the number in front of x)

      See if you can solve the equation. Comment back if you get stuck with what you have done so far.
      (11 votes)
  • aqualine seed style avatar for user David Burdak
    If I have one cookie and my friend eats my cookie is he really my friend?
    (5 votes)
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    • mr pink green style avatar for user David Severin
      It depends on a lot of circumstances; have you already eaten the rest of the bag and are offering him the last one?
      If you really wanted the cookie, did you give it away, or did they just take it?
      Why is there only one cookie?
      I do not feel we have enough information to answer this question.
      (5 votes)
  • cacteye yellow style avatar for user Evan Campbell
    This is still a little confusing for me
    (3 votes)
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  • blobby green style avatar for user Vanessa Mejia Gonzalez
    When all the numbers are negative the answer is negative always or the most times?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      It depends on what you are doing.
      If you are multiplying/dividing two negatives (or any even number of negatives), then the answer is positive. So if you have -2 x = -6, dividing by -2 gives x = -6/-2=3.
      If you are adding two negatives, the answer would always be a negative.
      (2 votes)
  • hopper cool style avatar for user josyu972
    Why is peter so silly
    (2 votes)
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  • blobby green style avatar for user Richard Mones
    Why do you multiply by 3/5 on both sides and not jut divide 15 by 5/3 in problem 2 ?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      If you know how to divide fractions, they end up the same thing. You want to divide 15/(5/3), to divide fractions, you reciprocate the denominator and multiply, so you end up with 15*3/5. So if you have 5/3 x, it is easier to multiply by the reciprocal so that you do not have to go through the extra step of dividing by a fraction.
      (1 vote)