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Chapter M3
Algebra

Learning Objectives
•
•
•
•
Understand and use algebraic terms
Manipulate algebraic expressions
Solve simple linear equations (including transposition)
Solve simultaneous linear equations (including the
graphical technique)
• Solve business problems using simple algebra
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 1
Definitions

Algebra applies quantitative concepts to unknown
quantities represented by symbols.

A term is a part of an expression that is connected to
another term by a plus or minus sign.

A constant is a term whose value does not change.

A variable is a term that represents a quantity that
may have different values.

An expression is a combination of constants and
variables using arithmetic operations.
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 2
Definitions

A coefficient is a factor by which the rest
of a term is multiplied.

The degree of expression is the highest
exponent of any variable in the
expression.

An equation is a statement that two
expressions are equal.
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 3
Algebraic rules
Rule:

If an expression contains like terms, these terms
may be combined into a single term. Like terms are
terms that differ only in their numerical coefficient.
Constants may also be combined into a single
constant.
Example:
5 x and 3 x are like terms
5 x and - 3 x may be combined
5 x - 3x  2 x
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 4
Algebraic rules
Rule:

When an expression is contained in brackets, each
term within the brackets is multiplied by any
coefficient outside the brackets.
Example:
Consider the expression :
23 x  4 y  1
to remove the brackets
23 x   24 y   2 1  6 x  8 y  2
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 5
Algebraic rules
Rule:

To multiply one expression by another, multiply
each term of one expression by each term of the
other expression. The resulting expression is said
to be the product of the two expressions.
The product of the two expressions
Example:
3 x  2  and 2 x  1
3 x  2 2 x  1  3 x 2 x  1  22 x  1
 6x2  3x  4x  2
 6x2  x  2
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 6
Algebraic rules
Rule:

Any term may be transposed from one side of an
equation to the other. When the transposition is
made, the sign of the term must change from its
original sign. ‘+’ becomes a ‘-’ and ‘-’ becomes a ‘+’.
Example:
15x - 20 = 12 - 4x
15x - 20 + 4x = 12
15x + 4x = 12 + 20
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 7
Solving linear equations
Solve
1.
9x - 27 = 4x + 3 for x
Place like terms of the variable on the left side of
the equation and the constant terms on the right
side.
9x - 4x = 3 + 27
2.
Collect like terms and constant terms.
5x = 30
3.
Divide both sides of the equation by the
coefficient of the variable (in this case 5).
x=6
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 8
Solving simultaneous equations
(finding the value of two variables)

Solve for x and y
3x + 4y = 33 (1)
2x - 3y = 5 (2)
Step 1:
Make the coefficient of either of the variables in one
equation equal to its coefficient in the other equation.
Multiply both sides of equation (1) by 2 and equation
(2) by 3.
6x + 8y = 66 (3)
6x - 9y = 15 (4)
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 9
Solve for x and y: 3x + 4y = 33 (1)
2x - 3y = 5 (2)
Step 2:
Eliminate the variable that has the same coefficient by:
subtracting equation (4) from equation (3). If the signs
are the same, then subtract equations. If the signs are
different, then add equations.
6x + 8y = 66 (3)
minus:6x - 9y = 15 (4)
equals 8y-(-9y) = 51
17y = 51
Divide both sides by 17 to find y.
y = 3
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 10
Solve for x and y: 3x + 4y = 33 (1)
2x - 3y = 5 (2)
Substitute in 3 for y in equation (1) and solve for x.
3x + 4(3) = 33
3x + 12 = 33
3x
= 33 - 12
3x
= 21
Divide both sides by 3.
x =7
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 11