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

Practice Problems: Three Step Linear Equations

The solution to each of the following problems is 20. Focus on finding the most helpful three or four algebraic steps to take someone reading your work from the problem as stated to the solution.

  1.   3(x+10)~=~90
  2.   2x-20~=~60-2x
  3.   \dfrac{2x}{3}~=~\dfrac{60-x}{3}
  4.   50~=~4(x-5)-10
  5.   x~=~\dfrac{100-2x}{3}
  6.   25-\dfrac{1}{4}x~=~x
  7.   2(x-10)-5~=~15
  8.   \dfrac{3x}{2}-5~=~x+5
  9.   4x-5~=~95-x
  10.   12~=~\dfrac{4x-20}{5}
  11.   -\dfrac{x+6}{2}~=~7-x
  12.   -x~=~4-\dfrac{6}{5}x
  13.   9-\dfrac{x}{2}~=~-1

    – Read each of the following, and decide on a variable you will use to answer the question.
    – Then write an algebraic equation that describes the situation using the variable you chose.
    – Then solve for the variable to answer the question:

  14. If Jimena has sixty three manga books, which is three times nineteen less than twice what Derrick has, how many manga books does Derrick have?
  15. Three friends collect athletic shoes. Esteban and Zuri have both collected the same number of athletic shoes. Esteban has ten fewer than three times Duong’s collection, while Zuri’s collection size is one hundred thirty pairs diminished by four times Duong’s collection. How many pairs are in Duong’s collection?

Related postings:
Practice: One Step Linear Equation Problems
– Practice: Two Step Linear Equation Problems
Practice: Ugly Linear Equation Problems

Practice Problems: Two Step Linear Equations

The solution to each of the following problems is 18. Focus on finding the most helpful algebraic steps to take a reader from the problem as stated to the solution.

  1.   2x+1~=~37
  2.   5+3x~=~59
  3.   2x+3x~=~90
  4.   2x~=~-4x+108
  5.   54~=~2x+x
  6.   92-20~=~4x
  7.   2(x-7)~=~22
  8.   39~=~(-5+x)\cdot3
  9.   33-15~=~10x-9x
  10.   12~=~\dfrac{2x}{3}
  11.   10~=~4+\dfrac{x}{3}
  12.   \dfrac{x+6}{4}~=~6

    – Read each of the following, and decide on a variable you will use to answer the question.
    – Then write an algebraic equation that describes the situation using the variable you chose.
    – Then solve for the variable to answer the question:

  13. Hector has 20 songs on his phone, which matches what Thea gets if she doubles the number of songs on her phone and reduces the result by 16. How many songs does Thea have on her phone?
  14. Aleena has two photographs in her room, which is a tenth of two more than the number Leon has in his room. How many photographs does Leon have in his room?
  15. Pedro has just started collecting Pokemon cards, while Amira has already collected 66 of them. If Pedro’s card count is increased by four, and this quantity is tripled, he will have as many as Amira. How many Pokemon cards does Pedro have?

Related postings:
Practice: One Step Linear Equation Problems
– Practice: Three Step Linear Equation Problems
Practice: Ugly Linear Equation Problems

Practice Problems: One Step Linear Equations

The solution to each of the following twenty problems is 12. Focus on finding the most helpful algebraic step to take a reader from the problem as stated to the solution, and be sure you can explain why that step leads to a solution.

  1.   x+7~=~19
  2.   3+x~=~15
  3.   11~=~-1+x
  4.   3x~=~12+2x
  5.   x-2~=~10
  6.   -18+x~=~-6
  7.   9~=~x-3
  8.   -5x~=~-6x+12
  9.   2x~=~24
  10.   84~=~7x
  11.   x\cdot 5~=~60
  12.   36~=~x\cdot 3
  13.   \dfrac{x}{3}~=~4
  14.   6~=~\dfrac{x}{2}
  15.   \dfrac{1}{4}x~=~3
  16.   \dfrac{4}{3}~=~x \cdot \dfrac{1}{9}

    – Read each of the following, and decide on a variable you will use to answer the question.
    – Then write an algebraic equation that describes the situation using the variable you chose.
    – Then solve for the variable to answer the question:

  17. Alicia is 15 years old. She is 3 years older than her younger brother Tony. How old is Tony?
  18. Mahmoud has 24 apples. He has twice as many as Janice. How many apples does Janice have?
  19. Gabriella’s immediate family has 8 family members. She has 4 fewer than Darius. How many immediate family members does Darius have?
  20. Lawrence has 6 books. He has half as many as Aisha. How many books does Aisha have?

Related postings:
Practice: Two Step Linear Equation Problems
– Practice: Three Step Linear Equation Problems
Practice: Ugly Linear Equation Problems

Interactive Graphs for Linear, Quadratic, Rational, and Trig Functions Moved to GeoGebraTube

Some may have had trouble using my GeoGebra applets in their browsers. I have moved all of them to GeoGebraTube, which will hopefully fix the problem. You may search for them by typing “MathMaine” into the GeoGebraTube search box.

Links to all updated interactive graph applets are below. Comments and suggestions are always welcome!

Linear Functions

GeoGebraBook: Exploring Linear Functions,
which contains:

Interactive Linear Function Graph: Slope-Intercept Form

Interactive Linear Function Graph: Point-Slope Form

Continue reading Interactive Graphs for Linear, Quadratic, Rational, and Trig Functions Moved to GeoGebraTube

Analyzing Linear Equations: a summary

Towards the end of the unit(s) on Linear Equations and their graphs, students can feel a bit overwhelmed.  The following is an attempt to summarize and link the key concepts you need to be comfortable with.

Lines

What is the least amount of information you need to know in order to be able to identify a line exactly? Two pieces of information: a point that the line passes through, as well as either a second point on the line or the line’s slope.

Continue reading Analyzing Linear Equations: a summary