Chapter 2: Linear Equations

2.1 Elementary Linear Equations

Solving linear equations is an important and fundamental skill in algebra. In algebra, there are often problems in which the answer is known, but the variable part of the problem is missing. To find this missing variable, it is necessary to follow a series of steps that result in the variable equalling some solution.

Addition and Subtraction Problems

To solve equations, the general rule is to do the opposite of the order of operations. Consider the following.

Example 2.1.1

Solve for [latex]x.[/latex]

  1. [latex]x-7=5[/latex]
    [latex]\begin{array}{rrrrr} x&-&7&=&-5\\ &+&7&&+7\\ \hline &&x&=&2 \end{array}[/latex]
  2. [latex]4+x=8[/latex]
    [latex]\begin{array}{rrrrr} 4&+&x&=&8\\ -4&&&&-4\\ \hline &&x&=&4 \end{array}[/latex]
  3. [latex]7=x-9[/latex]
    [latex]\begin{array}{rrrrr} 7&=&x&-&9\\ +9&&&+&9\\ \hline 16&=&x&& \end{array}[/latex]
  4. [latex]5=8+x[/latex]
    [latex]\begin{array}{rrrrr} 5&=&8&+&x\\ -8&&-8&&\\ \hline -3&=&x&& \end{array}[/latex]

Multiplication Problems

In a multiplication problem, get rid of the coefficient in front of the variable by dividing both sides of the equation by that number. Consider the following examples.

Example 2.1.2

Solve for [latex]x.[/latex]

  1. [latex]\begin{array}[t]{rrl} 4x&=&20\\ \\ \dfrac {4x}{4}&=&\dfrac{20}{4}\\ \\ x&=&5 \end{array}[/latex]
  2. [latex]\begin{array}[t]{rrl} 8x&=&-24\\ \\ \dfrac {8x}{8}&=&\dfrac{-24}{8}\\ \\ x&=&-3 \end{array}[/latex]
  3. [latex]\begin{array}[t]{rrl} -4x&=&-20\\ \\ \dfrac {-4x}{-4}&=&\dfrac{-20}{-4}\\ \\ x&=&5 \end{array}[/latex]

Division Problems

In division problems, remove the denominator by multiplying both sides by it. Consider the following examples.

Example 2.1.3

Solve for [latex]x.[/latex]

  1. [latex]\begin{array}[t]{rrl} \dfrac{x}{-7}&=&-2\\ -7\left(\dfrac{x}{-7}\right)&=&(-2)-7 \\ x&=&14\end{array}[/latex]
  2. [latex]\begin{array}[t]{rrl} \dfrac{x}{8}&=&5\\ 8\left(\dfrac{x}{8}\right)&=&(5)8\\ x&=&40\end{array}[/latex]
  3. [latex]\begin{array}[t]{rrl} \dfrac{x}{-4}&=&9\\ -4\left(\dfrac{x}{-4}\right)&=&(9) -4\\ x&=&-36\end{array}[/latex]

Questions

For questions 1 to 28, solve each linear equation.

  1. [latex]v + 9 = 16[/latex]
  2. [latex]14 = b + 3[/latex]
  3. [latex]x - 11 = -16[/latex]
  4. [latex]-14 = x - 18[/latex]
  5. [latex]30 = a + 20[/latex]
  6. [latex]-1 + k = 5[/latex]
  7. [latex]x - 7 = -26[/latex]
  8. [latex]-13 + p = -19[/latex]
  9. [latex]13 = n - 5[/latex]
  10. [latex]22 = 16 + m[/latex]
  11. [latex]340 = -17x[/latex]
  12. [latex]4r = -28[/latex]
  13. [latex]{-9} = \dfrac{n}{12}[/latex]
  14. [latex]27 = 9b[/latex]
  15. [latex]20v = -160[/latex]
  16. [latex]-20x = -80[/latex]
  17. [latex]340 = 20n[/latex]
  18. [latex]12 = 8a[/latex]
  19. [latex]16x = 320[/latex]
  20. [latex]8k = -16[/latex]
  21. [latex]-16 + n = -13[/latex]
  22. [latex]-21 = x - 5[/latex]
  23. [latex]p-8 = -21[/latex]
  24. [latex]m - 4 = -13[/latex]
  25. [latex]\dfrac{r}{14} = \dfrac{5}{14}[/latex]
  26. [latex]\dfrac{n}{8} = {40}[/latex]
  27. [latex]20b = -200[/latex]
  28. [latex]-\dfrac{1}{3} = \dfrac{x}{12}[/latex]

Answer Key 2.1

Extra Reading and Instructional Videos

Article to read: New theory finds ‘traffic jams’ in jet stream cause abnormal weather patterns.

The abstract reads:

A study offers an explanation for a mysterious and sometimes deadly weather pattern in which the jet stream, the global air currents that circle the Earth, stalls out over a region. Much like highways, the jet stream has a capacity, researchers said, and when it’s exceeded, blockages form that are remarkably similar to traffic jams — and climate forecasters can use the same math to model them both.

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