Unit 5 sec 5.1 solutions of equations

Unit 5 sec 5.1 solutions of equations

20 January 2016

22:37

As you saw earlier, an equation consists of two expressions, with an equals sign between them. Here’s an example:

2(r + 3) = 5r − 6.

The expressions to the left and right of the equals sign in an equation are referred to as the left-hand side (LHS) and right-hand side (RHS),

respectively.

LHS = 2(r + 3) and RHS = 5r − 6.

This section is about equations that contain an unknown – a letter representing a number that you don’t know. For example, earlier you saw the equation

3

10N = 150,

where N is an unknown representing the number of children who applied to a school. It turned out that N = 500, because equation) is correct

when you substitute 500 for N:

3 × 500 = 150.

10

The process of finding the value of the unknown in an equation is called solving the equation, and the value found is called a solution of the equation. We also say that this value satisfies the equation.

Example 18 Checking a solution of an equation

Show that r = 4 is a solution of the equation

2(r + 3) = 5r − 6.

Solution

If r = 4, then

LHS = 2(4 + 3) = 2 × 7 = 14

and

RHS = 5 × 4 −6 = 20 − 6 = 14.

Since LHS = RHS, r = 4 is a solution.

When you check whether a number is a solution of an equation, you should set out your working in a similar way to Example 18. Evaluate the left and right-hand sides separately, and check whether each side gives the

same answer.

 If one side of the equation is a constant term, then you just need to evaluate the other side, and check whether you get the right number. You can set out your working in the way shown in Example 19 below.

Example 19 Checking a solution of another equation

Show that x = −2 is a solution of the equation

1

2 (x + 8) = 3.

Solution

If x = −2, then

LHS = 1 (−2 + 8) = 1 × 6 = 3 = RHS.

           2                2

Hence x = −2 is a solution.

It’s possible for an equation to have more than one solution. For example, the equation a2 = 4 has two solutions, a = 2 and a = −2, because 22 = 4

and (−2)2 = 4. It’s also possible for an equation to have no solution at all. For example, the equation a2 = −1 has no solution, because squaring a real number always gives a non-negative answer.