1 Linear equations

Key Point 1

A linear equation is an equation of the form

a x + b = 0 a ≠ 0

where a and b are known numbers and x represents an unknown quantity to be found.

In the equation a x + b = 0 , the number a is called the coefficient of x , and the number b is called the constant term .

The following are examples of linear equations

3 x + 4 = 0 , − 2 x + 3 = 0 , − 1 2 x − 3 = 0

Note that the unknown, x , appears only to the first power, that is as x , and not as x 2 , x , x 1 ∕ 2 etc. Linear equations often appear in a non-standard form, and also different letters are sometimes used for the unknown quantity. For example

2 x = x + 1 3 t − 7 = 17 , 13 = 3 z + 1 , 1 − 1 2 y = 3 2 α − 1.5 = 0

are all examples of linear equations. Where necessary the equations can be rearranged and written in the form a x + b = 0 . We will explain how to do this later in this Section.

Task!

Which of the following are linear equations and which are not linear?

  1. 3 x + 7 = 0 ,
  2. − 3 t + 17 = 0 ,
  3. 3 x 2 + 7 = 0 ,
  4. 5 p = 0

The equations which can be written in the form a x + b = 0 are linear.

  1. linear in x
  2. linear in t
  3. non-linear - quadratic in x
  4. linear in p , constant is zero

To solve a linear equation means to find the value of x that can be substituted into the equation so that the left-hand side equals the right-hand side. Any such value obtained is known as a solution or root of the equation and the value of x is said to satisfy the equation.

Example 1

Consider the linear equation 3 x − 2 = 10 .

  1. Check that x = 4 is a solution.
  2. Check that x = 2 is not a solution.
Solution
  1. To check that x = 4 is a solution we substitute the value for x and see if both sides of the equation are equal. Evaluating the left-hand side we find 3 ( 4 ) − 2 which equals 10, the same as the right-hand side. So, x = 4 is a solution. We say that x = 4 satisfies the equation.
  2. Substituting x = 2 into the left-hand side we find 3 ( 2 ) − 2 which equals 4. Clearly the left-hand side is not equal to 10 and so x = 2 is not a solution. The number x = 2 does not satisfy the equation.
Task!

Test which of the given values are solutions of the equation

18 − 4 x = 26

  1. x = 2 ,
  2. x = − 2 ,
  3. x = 8
  1. Substituting x = 2 , the left-hand side equals

    18 − 4 × 2 = 10. But 10 ≠ 26 so x = 2 is not a solution.

  2. Substituting x = − 2 , the left-hand side equals:

    18 − 4 ( − 2 ) = 26 . This is the same as the right-hand side, so x = − 2 is a solution.

  3. Substituting x = 8 , the left-hand side equals:

    18 − 4 ( 8 ) = − 14 . But − 14 ≠ 26 and so x = 8 is not a solution.

Exercises
    1. Write down the general form of a linear equation.
    2. Explain what is meant by the root or solution of a linear equation.

In questions 2-8 verify that the given value is a solution of the given equation.

  1. 3 z − 7 = − 28 , z = − 7
  2. 8 x − 3 = − 11 , x = − 1
  3. 2 s + 3 = 4 , s = 1 2
  4. 1 3 x + 4 3 = 2 , x = 2
  5. 7 t + 7 = 7 , t = 0
  6. 11 x − 1 = 10 , x = 1
  7. 0.01 t − 1 = 0 , t = 100 .
    1. The general form is a x + b = 0 where a and b are known numbers and x represents the unknown quantity.
    2. A root is a value for the unknown which satisfies the equation.