1. MODULE 4
EQUATIONS
AND
INEQUALITIES

2. Solve the following equations: (a)
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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)
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3. Grade 10 Module 4 Page 60
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4. Grade 10 Module 4 Page 60
(b)
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5. EXAMPLE 2 Solve for x: "Textbook, Chapter, Page"
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EXAMPLE 2
Solve for x:
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7. VARIABLES IN THE DENOMINATOR
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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.
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8. EXAMPLE 3 (a) Solve for x: "Textbook, Chapter, Page"
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EXAMPLE 3
(a) Solve for x:
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9. But since the equation has no solution.
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But since the equation has no solution.
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(b) Solve for x:
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11. Grade 10 Module 4 Page 62
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valid solution

12. Grade 10 Module 4 Page 62
(c) Solve for x:
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15. Consider the following true statement:
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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
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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.
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17. Solve the following and then represent the solutions on a number line:
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EXAMPLE 4
Solve the following and then represent the solutions on a number line:
(a)
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19. Grade 10 Module 4 Page 63
(b)
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21. Grade 10 Module 4 Page 63
(c)
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22. Grade 10 Module 4 Page 64
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23. SIMULTANEOUS LINEAR EQUATIONS
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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:
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24. The values and are said to satisfy the equations simultaneously.
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True statement
True statement
The values and are said to satisfy the equations simultaneously.
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25. THE METHOD OF SUBSTITUTION EXAMPLE 5
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THE METHOD OF SUBSTITUTION
EXAMPLE 5
Solve for x and y: A
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26. Pick an equation and solve for one of the variables:
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Method 1 A B
Pick an equation and solve for one of the variables:
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27. Substitute into the other equation
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Substitute into the other equation
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28. Substitute into the equation
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Substitute into the equation
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30. Let’s solve for x in equation B:
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Method 2 A B
Let’s solve for x in equation B: C
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31. Now replace the variable x in equation A with and solve for y:
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Now replace the variable x in equation A with and solve for y:
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32. Now substitute into either equation A, B or C to get x:
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Now substitute into either equation A, B or C to get x:
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33. 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:
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34. Replace the variable x in equation B with
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Replace the variable x in equation B with
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36. THE METHOD OF ELIMINATION EXAMPLE 6 Solve for x and y:
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THE METHOD OF ELIMINATION
EXAMPLE 6
Solve for x and y:
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37. Method 1 A B Add Substitute into A or B "Textbook, Chapter, Page"
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Add
Substitute into A or B
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38. Grade 10 Module 4 Page 67
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39. Multiply each term of B by A
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Method 2 A B
Multiply each term of B by A B
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40. A C Substitute into A "Textbook, Chapter, Page"
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Substitute into A
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41. Grade 10 Module 4 Page 68
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42. Multiply each term of B by
Example 7
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Solve for x and y: A B
Method 1:
Multiply each term of B by A
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43. A C Add Substitute into A or B "Textbook, Chapter, Page"
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Add
Substitute into A or B
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44. Grade 10 Module 4 Page 68,69
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45. Multiply each term of A by Multiply each term of B by
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Method 2: A B
Multiply each term of A by
Multiply each term of B by C
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46. Grade 10 Module 4 Page 68,69 C D
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47. A quadratic equation has the form
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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
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48. For example, the zero factor law can be applied to the equation
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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:
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49. Solve each of the following equations: (a)
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EXAMPLE 8
Solve each of the following equations:
(a)
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(b)
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51. 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.
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53. Grade 10 Module 4 Page 71
(c)
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54. You may write the solution as This equation has two equal solutions.
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(d)
You may write the solution as
This equation has two equal solutions.
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(e)
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(f)
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58. Grade 10 Module 4 Page 71
But
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59. LITERAL EQUATIONS A literal equation is one in which letters
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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).
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60. (a) Make a the subject of the formula
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EXAMPLE 9
(a) Make a the subject of the formula
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61. Make the subject of the formula.
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In Physical Science, a formula which relates pressure, volume and temperature is given by the formula:
Make the subject of the formula.
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64. (b) The area of a circle is given by Make r the subject of the
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(b) The area of a circle is given by
Make r the subject of the
formula.
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