Linear equations | Lesson

What are linear equations?

• $x$‍ is the variable, which represents a number whose value we don't know yet.
• $2$‍ is the coefficient, or the constant multiple of the variable $x$‍. Together, $2x$‍ is a single
  term
  .
• $3$‍ and $4$‍ are constants. They are both terms as well.

What skills are tested?

• Solving linear equations with one variable

How do we solve linear equations?

• Addition and subtraction are inverse operations.
• Multiplication and division are inverse operations.

What features make linear equations more difficult?

Negative numbers

Fractions

• If the equation has only a fraction coefficient, consider leaving the fraction until the last step in isolating $x$‍.

  Example

  For the equation below, we leave the fraction until the last step:

  $\begin{array}{rl}{\displaystyle \frac{1}{2}}x+3& =5\\ \\ {\displaystyle \frac{1}{2}}x+3-3& =5-3\\ \\ {\displaystyle \frac{1}{2}}x& =2\\ \\ {\displaystyle \frac{1}{2}}x\times 2& =2\times 2\\ \\ x& =4\end{array}$‍
• If the equation has both fraction coefficients and fraction constants, consider getting rid of the fractions in the first step.

  Example

  For the equation below, we can get rid of the fractions in the first step by multiplying by the smallest common multiple of the denominators of the fractions:

  $\begin{array}{rl}{\displaystyle \frac{1}{2}}x+{\displaystyle \frac{1}{3}}& ={\displaystyle \frac{1}{5}}\\ \\ ({\displaystyle \frac{1}{2}}x+{\displaystyle \frac{1}{3}})\times 30& ={\displaystyle \frac{1}{5}}\times 30\\ \\ ({\displaystyle \frac{1}{2}}x\times 30)+({\displaystyle \frac{1}{3}}\times 30)& =6\\ \\ 15x+10& =6\\ \\ 15x+10-10& =6-10\\ \\ 15x& =-4\\ \\ {\displaystyle \frac{15x}{15}}& ={\displaystyle \frac{-4}{15}}\\ \\ x& =-{\displaystyle \frac{4}{15}}\end{array}$‍

More than one variable term

Coefficients to distribute

Your turn!

Things to remember

1. Distributing any coefficients.
2. Combining any like terms.
3. Isolating the variable.