2.1 Solving First – Degree Equations BobsMathClass.Com Copyright © 2010 All Rights Reserved. 1...

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2.1 Solving First – Degree Equations

BobsMathClass.Com Copyright © 2010 All Rights Reserved.

EquationsAn equation is a statement that two expressions are equal. Algebraic equations contain one or more variables.

2Examples: 2x 5 9, 3x 5y 15, and x 2x 48 0First-degree equations in one variable

A first-degree equations is an equation that contains only one variable and the variable has an exponent of 1 (Also called a linear equation).

x 2 3x 5Examples: 2x 5 9, 3x 4 2x 7 and

3 7

If a = b, then a + c = b + c. Addition property for equalityMultiplication prope

If arty o = b,f an equal then ity: ac =

: bc.

(for all real numbers a, b, and c, c 0)

To solve first-degree equations in one variable, we will need the following properties.

Procedure for Solving Equations of the form ax + b = cStep 1. Isolate the term with the variable by adding the opposite of the constant term, b,to both sides (Addition Property of Equality).Step 2. Multiply both sides by the reciprocal of ‘a’ on both sides (Multiplication property of equality).

Step 3. Simplify both sides of the equation.

Step 4. Write and Check the solution.

Note: There are two formats for solving equations, horizontal and vertical. The text shows only the horizontal format. This lesson will show both formats. Choose the method you are most comfortable with.

2



Isolate variable term by adding -2 to both sides.

4x + 2+(-2) 22+(-2)

4x 201

Multiply both sides by .4

Answer: x 5

4(5) 2 22Check: 4x 2 22 Replace x with 5.Simplify each side. 20 2 22

22 22

514

14

4x + 2 22 +(-2) +(-2)

Example 1. Solve and Check: 4x + 2 = 22.

Solution:(Horizontal Format)

Example 1. Solve and Check: 4x + 2 = 22.(Vertical Format)

Solution:

Answer: x 5

514

14

4x 20

Your Turn Problem #1

Solve and Check: 7x + 4 = 31

27 xAnswe

7r:

3



Add 2 to both sides.

1Multiply both sides by .

3

14 3( 4) 2 Simplify each side.

Check: 14 3a 2 Replace a with 4.

14 14 14 12 2

4 a

Answer: a 4 (Symmetric Property: if a=b , then b=a)

+2 -1

4 3a

2

+2

413

13

12 3a

Solution: Solve Example 14 32. a 2. Choose the method you are comfortable with. The

examples will usually use the vertical format.


Solve and check: 31 10c 11

Answer: c 2

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Procedure: Solving Equations of the form ax + b = cx + dStep 1. Add the opposite of the variable term on the right to both sides, then simplify.Step 2. Add the opposite of the constant term on the left to both sides, then simplify.Step 3. Multiply both sides by the reciprocal of the variable term (it should be on the left).Step 4. Write and Check the solution.

Solve and check Exa 5xmple 3 9a. 3 3x .

Solution:

(vertical format)

2x 3 9

12

12

2x 126

Answer: x 6

Check: 5x 3 3x 9

5( 6) 3 3( 6) 9 30 3 18 9

27 27

5x 3 3x 9 (-3x) +(-3x )

(-3) +(- 3)

5x 3 3x 9 (horizontal format)

( 3x)5x 3 3x 9(3x ) 2x 3 9

( 3)2 (3 )x 9 3

12

12

2x 12

Answer: x 6

6

Your Turn Problem #3Solve and Check: 7x 3 11x 3

Answer:3

x2

5



Equations may not always be written in the forms given on the previous slides. Therefore, always simplify both sides of the equation before performing the necessary steps.Recall the distributive p x(aroperty b): x xa b.

3(5x 2) 15x 6 , Exampl (4xes: 1) 4x 1

Solution:

Use the distributive property to get rid of parenthesis. Solve x 4 1Example 4. x 3 2x 3 8x.

x 4 4x 6x+9 8x 3x 4 2x 9

Proceed as in previous examples.2x 2xx 4 9

4 4x 13 Multiply by 1 on both sides

to get x by itself. 1 1

Answer: x 13

Check: x 4(1 x) 3(2x 3) 8x

13 4 1 ( 13) 3 2( 13) 3 8( 13)

13 4 12 3 26 3 8( 13) 13 48 3 23 8( 13)

35 69 104 35 35


Solve and Check: 56 2(3b 1) 3b 7

Answer: b 17

Then simplify left and right hand sides

6



Use the distributive property to get rid of parentheses.Solution:

Solve 7(x 2) (Example 3x 4) 3(2x 2) 6. 5.

4x 18 6x Simplify left and right hand side.

7x 14 3x 4 6x 6 6

Proceed as in previous examples.In this case, get x on the right hand side.

4x 4x18 2x

Answer: x 9

Check: 7(x 2) (3x 4) 3(2x 2) 6

7 ( 9) 2 3( 9) 4

7 11 23

77 23

54

3 2( 9) 2 6

3 18 2 6

3 16 6

48 6 54 54


3(4x 2) 2(Solve: x 6) 2(x 1)

1 Answ r: xe

2

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Procedure: Translating Sentences into Equations and SolvingStep 1. Assign a variable to the unknown quantity (What is the problem asking you to find?)Step 2. Translate the sentence into a mathematical expression.

Step 3. Solve the equation using the steps from the preceding slides.

Example 6. Translate and solve: If 8 is subtracted from 3 times a certain number, the result is 34. Find the number.

Solution: Let x represent the unknown number.

Recall: subtracted from translates to .a b b a

Translation: 8 subtracted from: 83 times a number: 3x

The result is 34: 34

3x 8 34 Translated:

Then solve:


If 3 is subtracted from twice a certain number,the result is 15. Find the number.

The number An isswer: 6.

8 8

3x 42x 14The number is 14.

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Note some facts from the examples above. Consecutive integers increase by one. Consecutive odd and even integers increase by two. The key to solving application problems is to follow the given “setups” for that kind of problem. Please follow all steps and procedures as shown. This will help you to obtain the equation and solution.

1. Let x be the first integer.

2. Let x+1 be the second integer.

3. Let x+2 be the third integer. Continue this process if asked to find more than three integers.4. Translate into a mathematical equation and solve.

5. Check the answer.

Procedure: To find the consecutive integers:

Consecutive Integer Problems

Integer Problems

The next type of application problems to consider are “consecutive integer” problems.

Consecutive even integers are integers that follow in a sequence where each number is 2 more that the previous number. Example of consecutive even integers: 12, 14, and 16.Consecutive odd integers are integers that also follow in a sequence where each number is 2 more that the previous number. Example of consecutive odd integers: -7, -5, and -3.

Consecutive integers are integers that follow in a sequence, each number is 1 more than the previous number. Example of consecutive integers: 17, 18, and 19.

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Find three consecutive integers whose sum is –36.

Example 7: Find three consecutive integers whose sum is -66.

Setup: First integer: xSecond integer: x+1Third integer: x+2

Translated:

x x 1 x 2 66

Solve:3x 3 66

3x 69x 23

Once x is found, the next step is to write the answer.Check:

23 ( 22) ( 21) 66

First integer: x =Second integer: x+1Third integer: x

2323 1 22

23 2 2= 1+2

Answer: The integers are –23, -22, and -21

11, 12,Answe andr: 13

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Procedure: To find consecutive even integers:1. Let x be the first even integer.2. Let x+2 be the second even integer.

3. Let x+4 be the third even integer. etc.

Consecutive Even Integers

4. Translate into a mathematical equation and solve.5. Check the answer.

Example 8. Find two consecutive even integers such that three times the first equals twice the second.Setup: First integer: x

Second integer: x+2

Translated: 3x 2(x 2)

Solve:3x 2x 4 x 4

Once x is found, the next step is to write the answer.

Check:

3(4) 2(6) 12=12

Three times the first equals twice the second.

First integer: x = Second integer: x+ 22

44 6

Answer: The integers are 4 and 6.

Your Turn Problem #8Find three consecutive even integers such that three times the middle integeris four more than the sum of the first and third.

Answer: 2, 4, and 6

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Procedure: To find consecutive odd integers:1. Let x be the first odd integer.2. Let x+2 be the second odd integer.

3. Let x+4 be the third odd integer. etc.

Consecutive Odd Integers

4. Translate into a mathematical equation and solve.5. Check the answer.

Example 9. Seven times the first of two consecutive odd integers is five times the second.

Find the two odd integers.Setup: First integer: xSecond integer: x+2

Translated: 7x 5(x 2)

Solve: 7x 5x 102x 10 x 5

Once x is found, the next step is to write the answer.

7(5) 5(7) 35=35

Check: Seven times the first equals twice the second.

First integer: x =Second integer: x+ 22

55 7

Answer: The integers are 5 and 7


Find three consecutive odd integers such that three times the middle integer is one more than the sum of the first and third.

The end.B.R.12-07-06Answer: -1, 1, 3

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