Practice Problems: Ugly Linear Equations

The title of this post reflects how I categorize problems. The solution to each of the following problems is 7. Focus on finding the most helpful series of algebraic steps to take someone reading your work from the problem as stated to the solution. As the problems begin to include more and more terms, be cautious about doing too much in any one step – as that is how errors often arise.

  1.   2(15-a)~=~4(a-3)
  2.   b-9+3b~=~10+5b+50-8b-20
  3.   3(m-1)-(m+3)~=~2(5-m)+(m+5)
  4.   \dfrac{4c-12+c+1}{2}~=~c+5
  5.   \dfrac{4d}{-3}~=~\dfrac{28}{6}-2d
  6.   \dfrac{x-7}{6}~=~\dfrac{7-x}{2}
  7.   \dfrac{11a-22}{11}+a-1~=~\dfrac{36-2a}{2}
  8.   w+\dfrac{w-16}{5}~=~\dfrac{3}{5}w+8-w
  9.   \dfrac{v-5}{2}-\dfrac{1}{4}~=~\dfrac{5}{2}-\dfrac{v}{4}
  10.   \dfrac{t+3}{2}+3t-\dfrac{10t+11}{7}~=~\dfrac{13t+10}{7}
  11.   \dfrac{-15+y}{6}+\dfrac{1-3y}{3}~=~-\dfrac{4(1+2y)}{3}+y+5
  12.   \dfrac{x-10}{5}+\dfrac{1}{7}~=~\dfrac{3-x}{7}+\dfrac{4}{35}
  13.   \dfrac{d+5}{4}+\dfrac{2}{3}d-1~=~\dfrac{87-d}{12}
  14.   \dfrac{2w-10}{5}-\dfrac{10-4w}{3}~=~6-\dfrac{4}{15}(w-10)
  15.   \dfrac{2(x+3)}{3}+2(\dfrac{x}{5}) -\dfrac{28}{7}~=~\dfrac{8}{3}+\dfrac{4x}{10}-\dfrac{2x-14}{12}
  16.   \dfrac{a+2}{7}+\dfrac{1}{10}(a+3)-\dfrac{3-a}{5}~=~\dfrac{a+2}{14}-\dfrac{2-4a}{35}+\dfrac{17}{70}a

By Whit Ford

Math tutor since 1992. Former math teacher, product manager, software developer, research analyst, etc.


  1. Thanks for the brain exercise! I had to approach some of the uglier ones a few times in order to come up with the correct answer, but knowing the final destination helped me to know when I was finished!

    1. I was rather embarrassed at how many errors I caught in the problems when I double-checked them before posting… throw in enough terms and negative signs, and it becomes pretty clear we are all human!

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