Maths revision Chapter 2.

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Chapter 2

An equation is a statement that two things are equal, such as:

2 + 2 = 4

7 x 5 = 30 + 5

4 + x = 10 + y

It consists of 2 expressions separated by the equal signs (=+, and it is considered true if the left hand side and right hand side are equal. True equations include 7 + 10 = 20 - 3 and 8 = 4 + 4; examples of false equations are 2 + 2 = 7 and 10 x 5 = 13.

If an equation has a pronumeral in it, such as 3 + x = 7, then a solution to the question is a value to substitute for the pronumeral to form a true equation. In this case, a solution is x = 4 because 3 + 4 = 7 is a true equation.

An equation is a mathematical statement that 2 expressions are equal, such as 4 + x =32. It could be true (eg. 4 +28 = 32) or false (eg. 4 + 29 = 32)

In an equation the pronumeral is sometimes is sometimes called an unknown.

A false equation can be made into a true statement by using the =/ sign. For instance, 4 + 2 =/ 32 is a true statement.

An equation has a left-hand-side (LHS) and a right-hand-side (RHS).

A solution to an equation is a value that makes an equation true. The process of finding a solution is called solving.

An equation could have no solutions or it could have one or more solutions.

If we have an equation, we can obtain an equivalent equation by performing the same operation to both sides. For example, if we have 2x + 4 = 20, we can add 3 to both sides to obtain 2x + 7 = 23.

The new equation will be true for exactly the same values of x as the old equation. This observation helps us to solve equations systematically. For example, 2x + 4 = 20 is equivalent to 2x = 16 (subtract 4 from both sides), and this is equivalent to x = 8 (divide both sides by 2). The bottom equation is only true if x has the value 8, so this means the solution to the equation 2x + 4 = 20 is x = 8.

Two equations are equivalent if you can get from one to the other by repeatedly

– adding a number to both sides

– subtracting a number from both sides

– multiplying both sides by a number other than zero

– dividing both sides by a number other than zero

– swapping the left-hand side and right-hand side of the equation.

To solve an equation systematically, you should repeatedly find an equivalent equation that is simpler. For example: 5x + 2 = 32: x = 6

To check a solution substitute the unknown’s value in to see if the equation is true, (e.g. LHS = 5(6) + 2 and RHS = 32).

So far all the equations we have considered involved a pronumeral either on the left-hand side,

e.g. 2x + 3 = 11 or on the right side, e.g. 15 = 10 − 2x. But how can you solve an equation with pronumerals on both sides, e.g. 12 + 5x = 16 + 3x? The idea is to look for an equivalent equation with pronumerals on just one side.

If both sides of an equation have a pronumeral added or subtracted, the new equation will be equivalent to the original equation.

If pronumerals are on both sides of an equation, add or subtract it so that the pronumeral appears on only one side. For example: (10 + 5a) = (13 + 2a); 10 + 3a = 13

Sometimes it is wise to swap the left-hand side and right-hand side.

Most of the equations you have worked with so far are called linear equations like 2x − 3 = 7 and 5(a + 2) = 7(a − 1) where the power of the pronumeral is 1 and there is usually a single solution. Another type of equation is of the form x2 = c and this is an example of a simple quadratic equation. Note that the power of the pronumeral x is 2. Depending on the value of c, x can have zero, one or two solutions. These types of equations appear frequently in mathematics and in problems involving distance, area, graphs and motion.

x^2 = c is a simple quadratic equation.

– If c>0, then there are two solutions: x= c and x=− c.

For example: The solutions for x^2 =16 are x=4 or x=−4(this can be written as x=±4)

because (−4)^2 = 16 and 4^2 = 16.

– If c=0, then there is one solution, x=0.

– If c < 0, then there are no solutions because x^2 will always be positive.

Some equations are called formulas. A formula shows the relationship between two or more variables. For example, you know from measurement that the area of a rectangle is related to its length and breadth, given by the formula A = l × b and its perimeter is given by P = 2l + 2b. Although these are often used as a definition for the area and a definition for the perimeter, they are also just equations − two expressions written on either side of an equals sign.

A formula or rule is an equation containing two or more pronumerals, one of which is the subject of the equation.

The subject of a formula is a pronumeral that occurs by itself on the left-hand side, e.g. V is the subject of V = 3x + 2y.

To use a formula substitute all the known values and then solve the equation to find the unknown value.

An equation can be used to describe any situation in which two values are equal. I To solve a problem follow these steps.

1 Define pronumerals to stand for unknown numbers (e.g. let j = John’s current age).

2 Write an equation to describe the situation.

3 Solve the equation systematically, if possible, or by inspection.

4 Ensure you answer the original question, including the correct units (e.g. dollars, years, cm).

5 Check that your solution is reasonable and makes sense.

An inequality is like an equation but, instead of indicating that two expressions are equal, it indicates which of the two has the greater value. For example, 2 + 4 < 7, 3 × 5 ≥ 15 and x ≤ 10 are all inequalities. The first two are true, and the last one could be true or false depending on the value of x. For instance, the numbers 9.8, 8.45, 7 and −120 all make this inequality true.

Given an inequality, an equivalent inequality can be obtained by:

– adding or subtracting an expression from both sides

– multiplying or dividing both sides by any positive number

– multiplying or dividing both sides by a negative number and reversing the inequality symbol

– swapping left-hand side and right-hand side and reversing the inequality symbol.