Everything Maths Grade 10

4 Equations and inequalities

4.1 Introduction EMA33

Equations are widely used to describe the world around us. In science equations are used to describe everything
from how a ball rolls down a slope to how the planets move around the sun.

In this chapter we will explore different types of equations as well as looking at how these can be used to solve
problems in the real world. We will also look at linear inequalities.

DID YOU KNOW?

The first use of an “equals” sign fromThe Whetstone of Witteby Robert Recorde 1557. This equation represents

14 x+15 = 71. Recorde is also responsible for introducing the pre-existing “plus” sign (+) to the English-speaking

world.

4.2 Solving linear equations EMA34

The simplest equation to solve is a linear equation. A linear equation is an equation where the highest exponent
of the variable is 1. The following are examples of linear equations:

2 x+ 2 = 1

2 x

3 x+ 1

= 2

4 (2x9) 4 x= 4 6 x

2 a 3

3

 3 a=

a

3

Solving an equation means finding the value of the variable that makes the equation true. For example, to solve
the simple equationx+ 1 = 1, we need to determine the value ofxthat will make the left hand side equal to
the right hand side. The solution isx= 0.

The solution, also called the root of an equation, is the value of the variable that satisfies the equation. For
linear equations, there is at most one solution for the equation.

To solve equations we use algebraic methods that include expanding expressions, grouping terms, and factoris-
ing.

For example:

2 x+ 2 = 1

2 x= 1 2 (rearrange)

2 x= 1 (simplify)

x=

1

2

(divide both sides by 2 )

74 4.1. Introduction