MODULE 4 EQUATIONS AND INEQUALITIES.

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Presentation transcript:

MODULE 4 EQUATIONS AND INEQUALITIES

Solve the following equations: (a) Grade 10 Module 4 Page 60 LINEAR EQUATIONS Linear equations have only one solution. For example, the only solution to the linear equation is since EXAMPLE 1 Solve the following equations: (a) "Textbook, Chapter, Page"

Grade 10 Module 4 Page 60 "Textbook, Chapter, Page"

Grade 10 Module 4 Page 60 (b) "Textbook, Chapter, Page"

EXAMPLE 2 Solve for x: "Textbook, Chapter, Page" Grade 10 Module 4 Page 60 EXAMPLE 2 Solve for x: "Textbook, Chapter, Page"

Grade 10 Module 4 Page 60 "Textbook, Chapter, Page"

VARIABLES IN THE DENOMINATOR Grade 10 Module 4 Page 61 VARIABLES IN THE DENOMINATOR In mathematics there is an extremely important principle: division by 0 is not permissible. Solving equations is all about finding a value for the variable so that the equality is true. You must check that your solutions won’t make any one of the denominators 0. That is the reason why we must state restrictions when solving equations. "Textbook, Chapter, Page"

EXAMPLE 3 (a) Solve for x: "Textbook, Chapter, Page" Grade 10 Module 4 Page 61 EXAMPLE 3 (a) Solve for x: "Textbook, Chapter, Page"

But since the equation has no solution. Grade 10 Module 4 Page 61 But since the equation has no solution. "Textbook, Chapter, Page"

Grade 10 Module 4 Page 62 (b) Solve for x: "Textbook, Chapter, Page"

Grade 10 Module 4 Page 62 "Textbook, Chapter, Page" valid solution

Grade 10 Module 4 Page 62 (c) Solve for x: "Textbook, Chapter, Page"

Grade 10 Module 4 Page 62 "Textbook, Chapter, Page"

Grade 10 Module 4 Page 62 "Textbook, Chapter, Page"

Consider the following true statement: Grade 10 Module 4 Page 63 LINEAR INEQUALITIES Consider the following true statement: If we now multiply (or divide) both sides by the statement will become which is clearly now false. However, if the direction of the inequality sign were reversed, then the statement would become true i.e. . "Textbook, Chapter, Page"

Grade 10 Module 4 Page 63 Therefore the following rules are always applicable when working with inequalities: • Change the direction of the inequality sign whenever you multiply or divide by a negative number. • Do not change the direction of the inequality if you multiply or divide by a positive number. "Textbook, Chapter, Page"

Solve the following and then represent the solutions on a number line: Grade 10 Module 4 Page 63 EXAMPLE 4 Solve the following and then represent the solutions on a number line: (a) "Textbook, Chapter, Page"

Grade 10 Module 4 Page 63 "Textbook, Chapter, Page"

Grade 10 Module 4 Page 63 (b) "Textbook, Chapter, Page"

Grade 10 Module 4 Page 63 "Textbook, Chapter, Page"

Grade 10 Module 4 Page 63 (c) "Textbook, Chapter, Page"

Grade 10 Module 4 Page 64 "Textbook, Chapter, Page"

SIMULTANEOUS LINEAR EQUATIONS Grade 10 Module 4 Page 65 SIMULTANEOUS LINEAR EQUATIONS It is sometimes necessary to solve two equations with two unknown variables. For example, the values of x and y that will satisfy the equations and are and . This can be checked by substituting these values into both equations as follows: "Textbook, Chapter, Page"

The values and are said to satisfy the equations simultaneously. Grade 10 Module 4 Page 65 True statement True statement The values and are said to satisfy the equations simultaneously. "Textbook, Chapter, Page"

THE METHOD OF SUBSTITUTION EXAMPLE 5 Grade 10 Module 4 Page 65 THE METHOD OF SUBSTITUTION EXAMPLE 5 Solve for x and y: A "Textbook, Chapter, Page" B

Pick an equation and solve for one of the variables: Grade 10 Module 4 Page 65 Method 1 A B Pick an equation and solve for one of the variables: "Textbook, Chapter, Page"

Substitute into the other equation Grade 10 Module 4 Page 65 Substitute into the other equation "Textbook, Chapter, Page"

Substitute into the equation Grade 10 Module 4 Page 65 Substitute into the equation "Textbook, Chapter, Page"

Grade 10 Module 4 Page 65 "Textbook, Chapter, Page"

Let’s solve for x in equation B: Grade 10 Module 4 Page 66 Method 2 A B Let’s solve for x in equation B: C "Textbook, Chapter, Page"

Now replace the variable x in equation A with and solve for y: Grade 10 Module 4 Page 66 Now replace the variable x in equation A with and solve for y: "Textbook, Chapter, Page"

Now substitute into either equation A, B or C to get x: Grade 10 Module 4 Page 66 Now substitute into either equation A, B or C to get x: "Textbook, Chapter, Page"

Grade 10 Module 4 Page 66 Method 3 A B Solve for x in equation A. The problem with method is that you will have to work with fractions, so it is not really recommended: "Textbook, Chapter, Page"

Replace the variable x in equation B with Grade 10 Module 4 Page 66 C Replace the variable x in equation B with "Textbook, Chapter, Page"

Grade 10 Module 4 Page 67 "Textbook, Chapter, Page"

THE METHOD OF ELIMINATION EXAMPLE 6 Solve for x and y: Grade 10 Module 4 Page 67 THE METHOD OF ELIMINATION EXAMPLE 6 Solve for x and y: "Textbook, Chapter, Page"

Method 1 A B Add Substitute into A or B "Textbook, Chapter, Page" Grade 10 Module 4 Page 67 A B Add Substitute into A or B "Textbook, Chapter, Page"

Grade 10 Module 4 Page 67 "Textbook, Chapter, Page"

Multiply each term of B by A Grade 10 Module 4 Page 68 Method 2 A B Multiply each term of B by A B "Textbook, Chapter, Page"

A C Substitute into A "Textbook, Chapter, Page" Grade 10 Module 4 Page 68 A C Substitute into A "Textbook, Chapter, Page"

Grade 10 Module 4 Page 68 "Textbook, Chapter, Page"

Multiply each term of B by Example 7 Grade 10 Module 4 Page 68,69 Solve for x and y: A B Method 1: Multiply each term of B by A "Textbook, Chapter, Page" C

A C Add Substitute into A or B "Textbook, Chapter, Page" Grade 10 Module 4 Page 68,69 A C Add Substitute into A or B "Textbook, Chapter, Page"

Grade 10 Module 4 Page 68,69 "Textbook, Chapter, Page"

Multiply each term of A by Multiply each term of B by Grade 10 Module 4 Page 69 Method 2: A B Multiply each term of A by Multiply each term of B by C "Textbook, Chapter, Page" D

Grade 10 Module 4 Page 68,69 C D "Textbook, Chapter, Page"

A quadratic equation has the form Grade 10 Module 4 Page 70 QUADRATIC EQUATIONS A quadratic equation has the form and has at most two real solutions. It is important to factorise the quadratic expression and then apply what is called the zero factor law, which states that if , then either or . "Textbook, Chapter, Page"

For example, the zero factor law can be applied to the equation Grade 10 Module 4 Page 70 This is due to the fact that if you multiply any number by zero the answer will always be zero. For example, the zero factor law can be applied to the equation as follows: "Textbook, Chapter, Page"

Solve each of the following equations: (a) Grade 10 Module 4 Page 70 EXAMPLE 8 Solve each of the following equations: (a) "Textbook, Chapter, Page"

Grade 10 Module 4 Page 70 (b) "Textbook, Chapter, Page"

Grade 10 Module 4 Page 70 Note: You could have first divided both sides by the number 3 in order to simplify the equation. However, you may never divide both sides by the variable you are solving for. The reason for this is that you will lose one of the solutions if you divide by the variable you are solving for. "Textbook, Chapter, Page"

Grade 10 Module 4 Page 71 "Textbook, Chapter, Page"

Grade 10 Module 4 Page 71 (c) "Textbook, Chapter, Page"

You may write the solution as This equation has two equal solutions. Grade 10 Module 4 Page 71 (d) You may write the solution as This equation has two equal solutions. "Textbook, Chapter, Page"

Grade 10 Module 4 Page 71 (e) "Textbook, Chapter, Page"

Grade 10 Module 4 Page 71 (f) "Textbook, Chapter, Page"

Grade 10 Module 4 Page 71 "Textbook, Chapter, Page"

Grade 10 Module 4 Page 71 But "Textbook, Chapter, Page"

LITERAL EQUATIONS A literal equation is one in which letters Grade 10 Module 4 Page 73 LITERAL EQUATIONS A literal equation is one in which letters of the alphabet are used as coefficients and constants. These equations, usually referred to formulae, are used a great deal in Science and Technology. The aim is to solve the equation or formula for a specific letter (or to make that letter the subject of the formula). "Textbook, Chapter, Page"

(a) Make a the subject of the formula Grade 10 Module 4 Page 73 EXAMPLE 9 (a) Make a the subject of the formula "Textbook, Chapter, Page"

Make the subject of the formula. Grade 10 Module 4 Page 73 In Physical Science, a formula which relates pressure, volume and temperature is given by the formula: Make the subject of the formula. "Textbook, Chapter, Page"

Grade 10 Module 4 Page 73 "Textbook, Chapter, Page"

Grade 10 Module 4 Page 73 "Textbook, Chapter, Page"

(b) The area of a circle is given by Make r the subject of the Grade 10 Module 4 Page 73 (b) The area of a circle is given by Make r the subject of the formula. "Textbook, Chapter, Page"