Grade 9 Linear Equations Mathematics Topic, Study notes of Mathematics

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MATH

ALGEBRA

LINEAR EQUATIONS

Week 1 Day 1 Linear Equations (Algebra and Trigonometry, Young 2 nd Edition, page 90-99)

GENERAL OBJECTIVE

• (^) Classify equations as linear, fractional, or rational,
• (^) Solve linear equations,
• (^) Solve equations leading to the form ax+b=0, and
• (^) Solve application problems involving linear equations by

developing mathematical models for real-life problems.

At the end of the lesson the students are expected to:

An equation is a statement that two mathematical expressions are

equivalent or equal.

DEFINITION EQUATION

The values of the unknown that makes the equation true are called

solutions or roots of the equation, and the process of finding the

solution is called solving the equation.

Example:

x 9

2

x  7  11 7  3 x  2  3 x

4 x 7 x 2  3 x 5

x 2

x

x 2

x 3

KINDS OF EQUATIONS

• (^) An identity equation is an equation that is true for any number

substituted to the variable.

. (x 1) 2 1 . ( 3 ) 3 . 3 4 4 3 2 2 2           c x x b x x x x a x x

Example:

• (^) Two equations with exactly the same solutions are called

equivalent equations.

. 4 . 5 2 22 . 5 20     c x b x a x

Example:

The following are equivalent equations.

• (^) An inconsistent equation is an equation that has no solution.
• (^) A consistent equation is an equation that has a solution.

For all real numbers a , b and c

1. Addition Property of Equality

If a = b then a + c = b + c

2. Subtraction Property of Equality

If a = b then a – c = b – c

3. Multiplication Property of Equality

If a = b then a ∙ c = b ∙ c c = b ∙ c = b ∙ c c

4. Division Property of Equality

If a =b then

wherec 0 c b c a   PROPERTIES OF EQUALITY

TODAY’S OBJECTIVE

• (^) Define linear equations in one variable,
• (^) Determine the difference between linear and nonlinear

equations,

• (^) Enumerate the steps in solving linear equations,
• (^) Solve linear equations and equations involving fractions,
• (^) Solve rational equations which are reducible to linear

equations,

• (^) Define extraneous solution.

At the end of the lesson the students are expected to:

DEFINITION LINEAR EQUATION IN ONE VARIABLE

A linear equation in one variable is an equation that can

be written in the form

a x + b = 0

where a and b are real numbers and a  0

Example:

2 x – 1 = 0, -5 x = 10 + x , 3 x + 8 = 2

Linear Equations Nonlinear Equations 4 x  5  3 2 8 2 x  x 

2 x  x

x  6 x  0

x

x  

 x

x

Nonlinear; contains the square of the variable Nonlinear; contains the reciprocal of the variable Nonlinear; contains the square root of the variable

EXAMPLE

STEP DESCRIPTION EXAMPLE

1 Simplify the algebraic expression on both sides 2(x-1)+3 = x-3(x+1) 2x-2+3 = x-3x- 2x+1 = -2x- 2 Gather all the variables on one side of the equation and all constant terms on the other side. 2x+2x = -3- 4x = - 3 Isolate the variable

x - 1

x

Problem #23 on page 97

Solve for the indicated variable: 2(x-1)+3=x-3(x+1)

Solve the following equations.

25 -  2 5y- 3  y 2  3  2 y 5   5  y 1  3 y 3 

pp. 97

46 -  7 - 8y 9  6y- 2  7  4 y 7  2  6  2 y 3  4 6 y

pp. 97

SOLVING RATIONAL EQUATIONS THAT ARE REDUCIBLE

TO LINEAR EQUATIONS

A rational equation is an equation that contains one or more

rational expressions.

Extraneous solution are solutions that satisfy a transformed

equation but do not satisfy the original equation.

Steps

1. Determine any excluded values(denominator equals 0).

2. Multiply the equation by the LCD.

3. Solve the resulting linear equation.

4. Eliminate any extraneous solution.

7 a

a

pp. 93 Classroom ex. 1.1.

1.   a(a 4 )

a

a- 4

pp. 94 Classroom ex. 1.1.



x 3 x

2 x 6

4x- 12

pp. 95 Classroom ex. 1.1.

2 

x 3

2x- 5

pp. 95 Classroom ex. 1.1.



Solve the following equations.

4 2 u 1 u u by Stewart,RedlinandWatson 2nd Edition Algebra&Trigonomet ry

5. exercise1.1 page 78           

EXAMPLE