Learn – Algebra foundations – Linear equations

Lesson notes

  1. You cannot solve one equation in two variables uniquely. You must express one variable in terms of the other variable using a second equation.
    • For example, if I say sum of two numbers x and y is 150, I could form the equation x+y = 150. But, I can have many possible values for x and y that satisfy this equation. For example, x = 100, y = 50 is one possibility, but x = 75, y = 75 is another possibility as well. And, x = 25, y = 125 is yet another possibility. To find the precise values of x and y, you need to have the equation in terms of one variable only. If we have additional information such as y is twice x, we can form another equation y = 2x, and replace y in the first equation by 2x.
  2. When you replace one variable by another variable, be sure to use parenthesis. Remember, PEMDAS rules!
    • For example, if I have an equation that says 2x + 3y = 18, I cannot solve it to get exact values of x and y. I need to have an equation in only one variable. Now, if I have a second equation that says x = y+4, I can replace x in the first equation by y+4. Now, if I rewrite the first equation as 2y+4+3y=18, this is wrong! Remember, in 2x+3y = 18, x is multiplied by 2. So, we should have the whole of (y+4) to be multiplied by 2. Thus, the correct way to write is using parentheses: 2(y+4)+3y = 18.

Equations in one variable

Exercises

Problem set

Solve for the variable in the following.

  1. a+99 = 80
  2. p+11 = -19
  3. x-8 = -60
  4. m-21 = -51

Problem set

Solve for the variable in the following.

  1. 3x = 21
  2. 4a = -20
  3. -p = 32
  4. -4m = 100
  5. -6r = -42
  6. -a = 21
  7. -x = -100

Problem set

Solve for the variable in the following.

  1. 2x+3 = 21
  2. 3a+7 = -20
  3. -5p+11 = -24

Problem set

Solve for the variable in the following.

  1. \frac{x}{3} = 10
  2. \frac{a}{5} = -2
  3. \frac{2p}{5} = 4
  4. \frac{3m}{4} = -30
  5. \frac{-3r}{2} = 15
  6. \frac{-2x}{3} = -10
  7. -\frac{7a}{5} = -35

Problem set

Solve for the variable in the following.

  1. \frac{5x}{6} = \frac{25}{18}
  2. \frac{3x}{7} = -\frac{33}{28}
  3. \frac{-8a}{7} = \frac{48}{49}
  4. \frac{-8p}{11} = -\frac{32}{33}
  5. \frac{-7n}{9} = -\frac{14}{27}
  6. -\frac{8b}{5} = \frac{16}{15}

Problem set

Solve for the variable in the following.

  1. 3a-7 = a+7
  2. y-8 = 8-y
  3. 6x+10 = 5+x
  4. -7p+3 = -27-2p
  5. -6-2m = 22+5m
  6. -5-7a = 31+2a
  7. 3(-b+2) = 2(b-12)
  8. 2(3w-50) = 3\left(35w+\frac{98}{3}\right)
  9. 4\left(2x-\frac{15}{2}\right) = 3\left(3x-\frac{11}{3}\right)
  10. 3\left(2x-\frac{7}{3}\right) = -5\left(\frac{13}{5}-\frac{3x}{5}\right)

Problem set

Solve for the variable in the following.

  1. 5\left(\frac{3}{5}a-2\right) = -7\left(-\frac{17a}{7}-\frac{1}{14}\right)
  2. 14\left(-\frac{1}{7}+\frac{3x}{2}\right) = -5\left(-\frac{3x}{5}-2\right)
  3. 2\left(\frac{5p}{6}-\frac{1}{3}\right)=3\left(\frac{p}{2}-\frac{1}{2}\right)
  4. 4\left(\frac{1}{10}-\frac{3x}{8}\right)=-3\left(\frac{x}{9}+\frac{1}{45}\right)
  5. -\frac{1}{2}(-4a)=-\frac{1}{3}(-45+9a)
  6. \left(-\frac{1}{2}+\frac{1}{3}\right)6p=2\left(p-\frac{1}{2}-\frac{1}{3}\right)
  7. \frac{1}{3}\left(6x-\frac{1}{4}-\frac{1}{5}\right)=\frac{1}{2}\left(6x-\frac{1}{2}+\frac{1}{5}\right)

Problem set

Solve for the variable in the following.

  1. \frac{x}{2} - 2 = \frac{5x}{4} - \frac{5}{2}
  2. 1-\frac{x}{5} = \frac{1}{3}-\frac{2x}{9}
  3. -3\left(\frac{2x}{7}-1\right) = -\frac{x}{2} + 8
  4. \frac{x}{2}-\frac{2x}{3}=\frac{x}{4}-\frac{x}{3}-2
  5. \frac{x}{2}-\frac{x}{3}=\frac{x}{4}-\frac{x}{5}

Problem set

Solve for the variable in the following.

  1. \frac{-2x}{3}+\left(\frac{-5}{4}\right)=-\left(-\frac{5}{6}+\frac{(-x)}{(-4)}\right)
  2. -3\left(\frac{3x}{2}+\left(-\frac{5}{15}\right)\right)=-2\left(\frac{2x}{3}+\frac{32}{12}\right)
  3. 2-\left(-\frac{9x}{18}-\frac{1}{9}\left(\frac{9x}{3}\right)\right)=\frac{1}{2}\left(\frac{x}{2}+\frac{8x}{6}\right)
  4. \frac{2x}{7}-\frac{1}{7}-\frac{1}{3}=\frac{x}{3}+\frac{14}{42}
  5. \frac{1}{5}+\left(\frac{4x}{-8}\right) = \frac{1}{-2}\left(\frac{16x}{5}-4-\frac{4}{5}\right)

Problem set

Solve for the variable in the following.

  1. \frac{a-1}{a+3} = 5
  2. \frac{x+1}{2x+1} = \frac{10}{11}
  3. \frac{1}{3}\left(\frac{3(x-2)}{2x+5}\right) = \frac{2}{3}
  4. \frac{10(2x-5)}{5} - \frac{6(x-2)}{6} = 4
  5. \frac{2x+5}{5} - \frac{2(3x+6)}{6} = 2
  6. \frac{4(x+3)}{12} + 14\left(\frac{2x-4}{7}\right) = 6

Problem set

Solve for the variable in the following.

  1. \frac{3}{x} = 22 - \frac{5}{2x}
  2. \frac{2}{3x} - \frac{5}{6x} = 1
  3. \frac{5}{3x-5}+\frac{7}{3x-5} = 3
  4. \frac{12}{3(2x+1)}-\frac{10}{2(2x+1)} = \frac{1}{3}
  5. \frac{1}{2(x+1)} - \frac{1}{3(x+1)} = 1
  6. \frac{6}{4(3-x)} + \frac{6}{9(3-x)} = -\frac{1}{6}
  7. \frac{6}{45(1-x)} - \frac{20}{24(1-x)} = \frac{2}{20}

Problem set

Solve for the variable in the following.

  1. \frac{3x-6}{2x-6} - \frac{5x-6}{2x-6} = \frac{1}{2}
  2. \frac{3x-5}{2x-5} - 3\left(\frac{x-\frac{11}{3}}{2x-5}\right) = 2
  3. \frac{3x+1}{2(4x-3)} - \frac{x+1}{3(4x-3)} = \frac{7}{14}
  4. \frac{15x-25}{19x-25} - \frac{45}{3}\left(\frac{x-6}{19x-25}\right) = 5
  5. \frac{x-1}{4(13x+20)} - \frac{x-1}{5(13x+20)} = \frac{1}{40}

Problem set

Solve for the variable in the following.

  1. \frac{3}{2(x+5)} - \frac{2}{3(x+5)} = \frac{5}{x}
  2. \frac{1}{2(2x+1)} - \frac{3}{4(1+2x)} = \frac{1}{x+5}
  3. \frac{2x-1}{7(2+x)} - \frac{9x-1}{7x+14} = \frac{1}{10}
  4. \frac{1}{3(1-x)} - \frac{1}{2(x-1)} = -\frac{1}{x}
  5. \frac{3x+3}{4(-2x+3)} - \frac{4x+3}{3(3-2x)} = \frac{1}{3}

Equations in two variables

Exercises

Problem set

Solve for both the variables in each of the following.

  1. p-q = 2 and p+q = 14
  2. b - a = 3 and 2a+b = 9
  3. x - y = 2 and 2y-x = -4
  4. m + 2n = 3 and 3m-n = -5
  5. p + q = 10 and 2q-p = -4

Problem set

Solve for both the variables in each of the following.

  1. x+y = 10 and 2x-3y = 10
  2. 3p+2q = 9 and 2p+3q = 1
  3. 5a-2b = 1 and 3b-2a = -7
  4. 2m-5n = 1 and 3m-4n = -2
  5. 3x+2y = -1 and 3y-4x = -10
  6. 5p+3q = -5 and 2q-3p = -16

Problem set

Solve for both the variables in each of the following.

  1. \frac{x}{y} = \frac{3}{7} and y-x = 16