BASIC MATHEMATICS FORM ONE-Topic 7: ALGEBRA

TOPIC 7: ALGEBRA

An algebraic expression – is a collection of numbers, variables, operators
and grouping symbols.Variables – are letters used to represent one or
more numbersAlgebraic OperationsSymbols to form Algebraic ExpressionsUse symbols to form algebraic expressionsThe parts of an expression collected together are called termsExample

• x + 2x – are called like terms because they have the same variables
• 5x +9y – are called unlike terms because they have different variables

An algebraic expression can be evaluated by replacing or substituting the numbers in the variablesExample 1Evaluate the expressions below, given that x = 2 and y = 3

Example 2Evaluate the expressions below, given that m = 1 and n = – 2

An expression can also be made from word problems by using letters and numbersExample 3A rectangle is 5 cm long and w cm wide. What is its area?SolutionLet the area be A.ThenA = length× widithA = 5w cm²Simplifying Algebraic ExpressionsSimplify algebraic expressionsAlgebraic expressions can be simplified by addition, subtraction, multiplication and divisionAddition and subtraction of algebraic expression is done by adding or subtracting the coefficients of the like terms or lettersCoefficient of the letter – is the number multiplying the letter<!–[endif]–>Multiplication and division of algebraic expression is done on the coefficients of both like and unlike terms or lettersExample 4Simplify the expressions below

Solution

Equations with One UnknownAn equation – is a statement that two expressions are equalAn Equation with One UnknownSolve an equation with one unknownAn equation can have one variable (unknown) on one side or two variables on both sides.When you shift a number or term from one side of equation to another, its sign changes

• If it is positive, it becomes negative
• If it is negative, it becomes positive

Example 5Solve the following equations

Solution

An Equation from Word ProblemsForm and solve an equation from word problemsSome word problems can be solved by using equations as shown in the below examplesExample 6Naomi is 5 years young than Mariana. The total of their ages 33 years. How old is Mariana?Solution

Mariana is 19 yearsEquations with Two UnknownsSimultaneous EquationsSolve simultaneous equationsSimultaneous equations – are groups of equations containing multiple variablesExample 7Examples of simultaneous equation

A simultaneous equation can be solved by using two methods:

• Elimination method
• Substitution method

ELIMINATION METHODSTEPS

• Choose a variable to eliminatee.g x or y
• Make
  sure that the letter to be eliminated has the same coefficient in both
  equations and if not, multiply the equations with appropriate numbers
  that will give the letter to be eliminated the same coefficient in both
  equations

• If the signs of the letter to be eliminated are the same, subtract the equations
• If the signs of the letter to be eliminated are different, add the equations

Example 8Solve the following simultaneous equations by elimination method

Solution

1. Eliminate y

To find y put x = 2 in either equation (i) or (ii)From equation (i)

(b)Eliminate x

In order to find y, put x = 2 in either equation (i) or (ii)From equation (ii)

(c) Given

To find g put r = 3 in either equation (i) or (ii)From equation (i)

(d) Given

To find x, put y = – 1 in either equation(i) or (ii)From equation (ii)

BY SUBSTITUTIONSTEPS

• Make the subject one letter in one of the two equation given

• Substitute the letter in the remaining equation and proceed as in case of elimination

Example 9Solve the following simultaneous equations by substitution method

Solution

Linear Simultaneous Equations from Practical SituationsSolve linear simultaneous equations from practical situations<!–[endif]–>Simultaneous equations can be used to solve problems in real life involving two variablesExample 10If
3 Mathematics books and 4 English books weighs 24 kg and 5 Mathematics
books and 2 English books weighs 20 kg, find the weight of one
Mathematics book and one English book.SolutionLet the weight of one Mathematics book = x andLet the weight of one English book = y

To find y, put x = 2.29 in either equation (i) or (ii)From equation(i).

InequalitiesAn
inequality – is a mathematical statement containing two expressions
which are not equal. One expression may be less or greater than the
other.The expressions are connected by the inequality symbols<,>,≤
or≥.Where< = less than,> = greater than,≤ = less or equal and ≥ =
greater or equal.Linear Inequalities with One UnknownSolve linear inequalities in one unknownAn
inequality can be solved by collecting like terms on one side.Addition
and subtraction of the terms in the inequality does not change the
direction of the inequality.Multiplication and division of the sides of
the inequality by a positive number does not change the direction of the
inequality.But multiplication and division of the sides of the
inequality by a negative number changes the direction of the inequalityExample 11Solve the following inequalities

Solution

Linear Inequalities from Practical SituationsForm linear inequalities from practical situationsTo represent an inequality on a number line, the following are important to be considered:

• The endpoint which is not included is marked with an empty circle
• The endpoint which is included is marked with a solid circle

Example 12

Compound statement – is a statement made up of two or more inequalitiesExample 13Solve the following compound inequalities and represent the answer on the number line

Solution