linear equation

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CoverageIn this chapter you will learn how to deal with:

equation containing bracket equation containing fraction equation containing decimals inequalities and solving inequalities problem solving involving two inequalities

An equation is a statement such as 3 x + 2 = 17 which contains an unknown number. In this case the unknown number is x.

The aim of this topic is to show how to find the value of the unknown number x.

All equations contain an 'equals' sign. To solve the equation, you need to reorganise it so that the unknown value is by itself on one side of the equation. This is done by performing operations on the equation.

When you do this, in order to keep the equality of the sides, you must remember that whatever you do to one side of an equation, you must also do the same to the other side. This is often called the balance method.

Some simple algebraic equations can be solved in just one step, or even by inspection. For example, if x + 2 = 9 you can spot, by inspection, that x must be 7.

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SECTION 3 : ALGEBRALINEAR EQUATIONS & INEQUALITIES

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A more formal approach would be to subtract 2 from both sides, so that x = 9 – 2 = 7.

In this section you will be solving simple equations of this kind. Although you may well be able to spot some of the solutions by inspection, it is better to solve them formally, since this equips you with the skills needed for harder equations where the solutions cannot be spotted by inspection.

A. LINEAR EQUATION (BASIC)

Example 1Solve the following linear equations.

a) 3x+ 7 = 22

SOLUTION

b) 11 - 5x = 26


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c)

d)


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QUESTION 1Solve for x:





EXERCISE A

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B. EQUATION WITH REPEATED UNKNOWN

Equations where the unknown appears more than once need to be solved systematically. Generally, we:

expand any brackets collect like terms use inverse operations to isolate the unknown while at the same

time maintaining the balance of the equation.

Example 2Solve the following linear equations

SOLUTION


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When the unknown appears on both sides of the equation, remove it from one side. Aim to do this so the unknown is left with a positive coefficient.


SOLUTION


SOLUTION


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EXERCISE B

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C. EQUATION WITH FRACTIONS


SOLUTION


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SOLUTION


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If the unknown appears as part of the denominator, we still solve by: writing the equations with the lowest common denominator

(LCD) and then equating numerators.


SOLUTION


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EXERCISE C

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D. FORMING EQUATION BASED ON PROBLEMS

Many problems we are given are stated in words. Before we can solve a worded problem, we need to translate the given statement into a mathematical equation.

We then solve the equation to find the solution to the problem. The following steps should be followed:


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Step 1: Decide what the unknown quantity is and choose a variable such as x to represent it.

Step 2: Look for the operation(s) involved in the problem. For example, consider the key words in the table below.

Step 3: Form the equation with an “=” sign. These phrase indicate equality: “the answer is”, “will be”, “the result is”, “is equal to”, or simply “is”

Example 8Translate into an equation:

a “When a number is added to 6, the result is 15.”b “Twice a certain number is 7 more than the number.”

SOLUTION


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With practice you will find that you can combine the steps, but you should note:

the mathematical sentence you form must be an accurate translation of the information for these types of problems, you must have only one variable in

your equation.

Example 9Translate into an equation: “The sum of 2 consecutive even integers is 34.”

SOLUTION

Example 10Apples cost 13 cents each and oranges cost 11 cents each. If I buy 5 more apples than oranges and the total cost of the apples and oranges is RM2.33, write alinear equation involving the total cost.

SOLUTION


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QUESTION 1Translate into linear equations, but do not solve:

a When a number is increased by 6, the answer is 13.b When a number is decreased by 5, the result is ¡4.


EXERCISE D

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c A number is doubled and 7 is added. The result is 1.d When a number is decreased by 1 and the resulting number is halved, the answer is 45.e Three times a number is equal to 17 minus the number.f Five times a number is 2 more than the number.

QUESTION 2Translate into linear equations, but do not solve:

a The sum of two consecutive integers is 33.b The sum of 3 consecutive integers is 102.c The sum of two consecutive odd integers is 52.d The sum of 3 consecutive odd integers is 69.

QUESTION 2Write an equation for each of the following:

a Peter is buying some outdoor furniture for his patio. Tables cost RM 40 each and chairs cost RM25 each. Peter buys 10 items of furniture at a total cost of RM 280. (Let the number of tables purchased be t.)

b Pencils cost 40 cent each and erasers cost 70 cent each. If I purchase three fewer erasers than pencils, the total cost will be RM3.40. (Let the variable p represent the number of pencils purchased.)

c A group of friends went to a cafe for tea and coffee. Tea costs RM 2.50 and coffee costs RM 3.60. The number of people who ordered coffee was twice the number who ordered tea, and the total bill was RM 29.10. (Let the number of people who ordered tea be q.)

E. FORMING EQUATION & SOLVE IT BASED ON PROBLEMS

Example 11


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The sum of 3 consecutive even integers is 132. Find the smallest integer.

SOLUTION

Example 12If twice a number is subtracted from 11, the result is 4 more than the number.What is the number?

SOLUTION


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Example 13Cans of sardines come in two sizes. Small cans cost $2 each and large cans cost $3 each. If 15 cans of sardines are bought for a total of $38, how many small cans were purchased?

SOLUTION

QUESTION 1Solve the following problem involving numbers.


EXERCISE E

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a) When a number is doubled and the result is increased by 6, the answer is 20. Find the number.

b) The sum of two consecutive integers is 75. Find the integers.

c) The sum of three consecutive even integers is 54. Find the largest of them.

d) When a number is subtracted from 40, the result is 14 more than the original number. Find the number.

e) When 22 is subtracted from a number and the result is doubled, the answer is 6 more than the original number. Find the number.

f) When one quarter of a number is subtracted from one third of the number, the result is 7. Find the number.

QUESTION 2Solve the following problem.

a) Roses cost $5 each and geraniums cost $3 each. Michelle bought 4 more geraniums than roses, and in total she spent $52. How many roses did she buy?

b) Nick has 40 coins in his collection, all of which are either 5-cent or 10-cent coins. If the total value of his coins is $3.15, how many of each coin type does he have?

c) A store sells batteries in packets of 6 or 10. In stock they have 25 packets which contain a total of 186 batteries. How many of each packet size are in stock?

F. POWER EQUATION


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QUESTION 1Solve for x, giving your answers correct to 3 significant figures:


G. INTERPETING LINEAR INEQUALITIES


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What is inequality ?

The speed limit when passing roadworks is often 25 kilometres per hour.This can be written as a linear inequality using the variable s to represent the speed of a car in km per h. s 25 reads ‘s is less than or equal to 25’.

Less than

Less than or equal

More than

More than or equal

We can also represent the allowable speeds on a number line:

The number line shows that any speed of 25 km per h or less is an acceptable speed.We say that these are solutions of the inequality.

Suppose our solution to an inequality is , so every number which is 4 or greater

than 4 is a possible value for x. We could represent this on a number line by:


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Likewise if our solution is x < 5 our representation would be:

Example 16Represent the following inequalities on a number line:

SOLUTION


EXERCISE G

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QUESTION 1Represent the following inequalities on a number line:

QUESTION 2Write down the inequality used to describe the set of numbers:

H. SOLVING LINEAR INEQUALITIES


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Example 17


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SOLUTION

Example 18


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SOLUTION

Example 19

SOLUTION

QUESTION 1


EXERCISE H

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Solve for x and graph the solutions:

QUESTION 2Solve for x and graph the solutions:

QUESTION 3Solve for x and graph the solutions

QUESTION 4Solve for x


linear equation

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