MYF2519
Algebra Unit 02
Session 1: Basic Algebra
Session 1
Basic Algebra
Session outline
1.1 Introduction
1.2 Laws on Squares and Laws Cubes
1.3 Solution of Two Simultaneous linear equations
1.4 Solutions of simultaneous a quadratic equation and a linear equation
1.5 Review Questions
1.1 Introduction
This session provides simplification techniques required to study Foundation Courses in
Mathematics.
Objectives
By the end of this session, you should be able to
• simplify algebraic expressions using Laws on squares and Laws on cubes.
● use direct method to find the solutions of two simultaneous linear equations.
● find the solutions of simultaneous quadratic equation and linear equations.
1.2 Laws on Squares and Laws on Cubes
1.2.1 Laws on Squares
For every
(a)
(b)
(c)
, we have
;
; and
.
Page
1
Proof: We have
Copyright @ 2021, The Open University of Sri Lanka
MYF2519
Algebra Unit 02
Session 1: Basic Algebra
1.2.2 Laws on Cubes
For every
(a)
(b)
, we have
;
.
Proof: We have
1.2.3 Activity Expand each of the following expressions:
(a)
(d)
(b)
(e)
(c)
(f)
(g)
(h)
(i)
(j)
(k)
1.2.4 Activity Simplify each of the following expressions:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(a)
(b)
Copyright @ 2021, The Open University of Sri Lanka
(c)
Page
2
1.2.5 Activity Simplify each of the following expressions:
MYF2519
Algebra Unit 02
(d)
Session 1: Basic Algebra
(e)
(g)
(f)
(h)
1.2.6 Activity Simplify each of the following expressions:
(a)
(b)
(d)
(e)
(g)
(h)
(c)
(f)
(i)
1.3 Solution of Two Simultaneous linear equations
Simultaneous equations are groups of equations containing two or more variables. In this
section, we look at pairs of linear equations involving the variables x and y.
Each equation, can be represented by a linear graph that is true for many x- and y-values.
If the graphs intersect, the values of x and y at the intersection are those that make both
equations true.
Both graphs have the
same x and y values here
x
Copyright @ 2021, The Open University of Sri Lanka
Page
3
y
MYF2519
Algebra Unit 02
Session 1: Basic Algebra
O
Consider a pair of simultaneous equations in two variables, of the type
.
The condition
given two lines are not parallel.
Now let us consider directly method to solve two simultaneous equations.
1.3.1 Direct method to solve two simultaneous equations
Suppose that we want to solve two equations
Let us first find the value of x.
D
N
The terms in the denominator can be obtained as follows.
Copyright @ 2021, The Open University of Sri Lanka
Page
Step 2 Multiply coefficient of x in the second equation and coefficient of y in the first
equation.
4
Step 1 Multiply coefficient of x in the first equation and coefficient of y in the second
equation.
MYF2519
Algebra Unit 02
Session 1: Basic Algebra
Step 3 Subtract the value you obtained in Step 2 from the value you obtained in Step 1.
Let this value be D.
The terms in the numerator can be obtained as follows.
Step 4 Multiply constant tem in the first equation and coefficient of y in the second
equation.
Step 5 Multiply constant tem in second equation and coefficient of y in the equation
of x in the first equation.
Step 6 Subtract the value you obtained in Step 5 from the value you obtained in Step 4.
Let this value be N.
Then
Similarly if we follow the similar procedure we can find the value of y and it is given by
D
N
.:
1.3.2 Example
Solve each of the following simultaneous equations
Page
5
Solution:
Copyright @ 2021, The Open University of Sri Lanka
MYF2519
Algebra Unit 02
Session 1: Basic Algebra
1.3.3 Activity Solve each of the following simultaneous equations
1.3.4 Activity Solve each of the following simultaneous equations
Copyright @ 2021, The Open University of Sri Lanka
Page
In previous studies you have dealt with pairs of simultaneous linear equations and
solved these using algebra. You may recall that the solution could also be represented
by the intersection of graphs of the equations and the equations solved by finding the
6
1.4 Solutions of simultaneous a quadratic equation and a linear equation
MYF2519
Algebra Unit 02
Session 1: Basic Algebra
coordinates of the point of intersection. The same is true when we have one linear and
one quadratic equation as a pair of simultaneous equations.
When solving a linear and quadratic equation simultaneously, a new quadratic
equation is formed, as you will see in the following examples.
1.4.1 Example Solve the system of equations
and
.
Solution:
Equate [1] and [2]. We have
and
In this case the straight line intersects the curve
points
and
.
Therefore the solutions are
1.4.2 Example Solve the system of equations
(parabola) at two distinct
and
.
Solution:
Equate [1] and [2]. We have
. In this case the straight line touches the curve
(parabola) at the point
.
Therefore the solution is
1.4.3 Example 3 Solve the system of equations
and
.
Solution :
Page
7
Equate (1) and (2). We have
Copyright @ 2021, The Open University of Sri Lanka
MYF2519
Algebra Unit 02
Session 1: Basic Algebra
In this case the straight line does not intersect the curve
( parabola).
1.5 Review Questions
1. Simplify each of the following expressions:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
2 . Solve the following system of equations
3. Solve the following system of equations
(a)
(c)
(e)
and
and
and
. (b)
. (d)
.
and
and
(f)
.
.
and
Page
8
.
Copyright @ 2021, The Open University of Sri Lanka
MYF2519
Algebra Unit 02
Session 1: Basic Algebra
4. Find the intersection points (if any exists) of each of the following graphs.
(a)
(c)
and
and
. (b)
.
and
(d)
.
and
Page
9
.
Copyright @ 2021, The Open University of Sri Lanka