Simplifying Algebraic Expressions and Equations of Equations and Inequalities s v6 Xzn s1

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Simplifying Algebraic

Expressions and Equations

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www.ck12.org Concept 1. Simplifying Algebraic Expressions and Equations

CONCEPT 1 Simplifying AlgebraicExpressions and Equations

ObjectiveTo solve and simplify algebraic expressions and equations for a variable or number.

Review Queue

Use the Order of Operations to simplify the following expressions.

1. 9 + 6 2 + 6245

2. (9 + 6) 2 + 62 (45)

3. 7(3x4) +x

4. x + 5 (4x + 3)

Evaluating Algebraic Expressions and Equations

Objective

To solve an expression or equation for a given value of a variable.

Watch This

MEDIA

Click image to the left for more content.

Khan Academy: Variables Expressions and Equations

Guidance

In #3 and #4 in the Review Queue above, we introduce a letter into a mathematical expression. These letters, also

called variables,represent an unknown number. One of the goals of algebra is to solve various equations for avariable. Typically,x is used to represent the unknown number, but any letter can be used.

To evaluate an expression or equation, that means, a value for the variable will be given and we need to plug it into

the expression or equation and test it. In order for the given value to be true for an equation, the two sides of the

equation must simplify to the same number.

Example A

Evaluate 2x29 forx= 3.

Solution:We know that 2x29 is an expression because it does not have an equals sign. Therefore, to evaluate this

expression, plug in -3 forx and simplify using the Order of Operations.

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2(3)29 (3)2 = 3 3=9

2(9)9

189

9

You will need to remember that when squaring a negative number, the answer will always be positive. There are

three different ways to write multiplication: 29,2 9, and 2(9).

Example B

Determine ifx=5 is a solution to 3x11=14.

Solution:Even though the directions are different, this problem is almost identical to Example A. However, this is

an equation because of the equals sign. Both sides of an equation must be equal to each other in order for it to be

true. Plug in 5 everywhere there is anx. Then, determine if both sides are the same.

?

3(5)11=14

1511 =14

4 =14

Because 4 =14, this is not a true equation. Therefore, 5 is not a solution.

Example C

Determine ift= 2 is a solution to 7t29t10=36.

Solution:Here,tis the variable and it is listed twice in this equation. Plug in -2 everywhere there is a tand simplify.

?

7(2)29(2)10=36

?

7(4) + 1810=36

?

28 + 1810=36

36=36

-2 is a solution to this equation.

Guided Practice

1. Evaluates3 5s + 6 fors=4.

2. Determine ifa= 1 is a solution to 4aa2 + 11= 22a.

Answers

1. Plug in 4 everywhere there is an s.

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10. x2 5x + 4=0;x=411. y37=y + 3;x=212. 7x3=4;x=113. 6z +z5=2z + 12;z= 314. 2b5b2 + 1=b2; b=615. 1

4g + 9=g + 15; g= 8

Find the value of each expression, given that a= 1,

b=2,

c= 4, andd=0.

16. ab c

17. b2 + 2d18. c + 1

2ba

19. b(a + c)d2

For problems 20-25, use the equationy2 +y12=0.

20. Isy=4 a solution to this equation?21. Isy= 4 a solution to this equation?22. Isy=3 a solution to this equation?23. Isy= 3 a solution to this equation?24. Do you think there are any other solutions to this equation, other than the ones found above?

25. Challenge Using the solutions you found from problems 20-23, find the sum of these solutions and their

product. What do you notice?

Simplifying Algebraic Expressions

Objective

To combine like terms within an expression.

Watch This

MEDIA


James Sousa: Combining Like Terms

Guidance

You might have noticed from the previous concept, that sometimes variables and numbers can be repeated within an

expression. If the same variable is in an expression more than once, they can be combined by addition or subtraction.

This process is calledcombining like terms.

Example A

Simplify 5x123x + 4.

Solution:Reorganize the expression to group together thexs and the numbers. You can either place the like terms

next to each together or place parenthesis around the like terms.

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5x123x + 4

5x3x12 + 4or(5x3x) + (12 + 4)

2x8

Notice that the Greatest Common Factor (GCF) for 2x and 8 is 2. Therefore, you can and use the Distributive

Property to pull out the GCF to get 2(x4).

Example B

Simplify 6a5b + 2a10b + 7

Solution: Here there are two different variables, a and b. Even though they are both variables, they aredifferent

variables and cannot be combined. Group together the like terms.

6a5b + 2a10b + 7

(6a + 2a) + (5b10b) + 7

(8a15b + 7)

There is only one number term, called the constant, so we leave it at the end. Also, in general, list the variables in

alphabetical order.

Example C

Simplifyw2 + 94w2 + 3w47w11.

Solution: Here we have one variable, but there are different powers (exponents). Like terms must have the same

exponent in order to combine them.

w2

+ 94w2

+ 3w4

7w113w4 + (w24w2)7w + (911)

3w43w27w2

When writing an expression with different powers, list the powers from greatest to least, like above.

Guided Practice

Simplify the expressions below.

1. 6s7t+ 12t10s

2. 7y29x2 +y214x + 3x24

Answers

1. Combine thess and thets.

6s7t+ 12t10s

(6s10s) + (7t+ 12t)

4s + 5t

2. Group together the like terms.

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7y29x2 +y214x + 3x24

(9x2 + 3x2) + (7y2 +y2)14x4

6x2 + 8y214x4

Notice in #1, we did not write(6s10s) (7t+ 12t)in the second step. This would lead us to an incorrect answer.

Whenever grouping together like terms and one is negative (or being subtracted), always change the operator toaddition and make the number negative.

In #2, we can also take out the Greatest Common Factor of -2 from each term using the Distributive Property. This

would reduce to 2(3x24y2 + 7x + 2). In this case, we take out a -2 so that the first term is positive.

Vocabulary

Constant

A number that is added or subtracted within an expression. In the expression 3x28x15, -15 is the constant.

Greatest Common Factor (GCF)

The largest number or variable that goes into a set of numbers.

Problem Set

Simplify the following expressions as much as possible. If the expression cannot be simplified, write cannot be

simplified.

1. 5b15b + 8d+ 7d2. 611c + 5c183. 3g27g2 + 9 + 124. 8u2 + 5u3u29u + 145. 2a5f

6. 7pp2 + 9p + q2165q2 + 67. 20x613x + 198. 8n25n2 + 9n + 14

Find the GCF of the following expressions and use the Distributive Property to simplify each one.

9. 6a18

10. 9x215

11. 14d+ 712. 3x24y + 21

ChallengeWe can also use the Distributive Property and GCF to pull out common variables from an expression.

Find the GCF and use the Distributive Property to simplify the following expressions.

13. 2b25b

14. m3 6m2 + 11m15. 4y4 12y38y2

Solving Algebraic Equations for a Variable

Objective

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To isolate the variable in an equation or formula.

Watch This

MEDIA


Khan Academy: Example of Solving for a Variable

Guidance

Solving an algebraic equation for a variable can be tricky, but it can also be very useful. This technique can be

used to go back and forth between to different units of measurement. To solve for, or isolate, a variable within an

equation, you must undo the operations that are in the equation.

Example A

Solve forx in the equation 3x4y=12.

Solution:To solve forx, you need to move the 4yto the other side of the equation. In order to do this, we must do

the opposite operation. Since the 4yis being subtracted, we must add it to the other side of the equation.

3x4y=12

+4y= +4y

3x=4y + 12

Now we need to get x alone. 3xis the same as 3 multiplied by x. Therefore, to undo the multiplication we mustdivide both sides of the equation by 3.

3x

3 =

4y

3 +

12

3

x=4

3y + 4

When undoing multiplication, you must divideeverythingin the equation by 3.

A few things to note:

1. To undo an operation within an equation, do the opposite.

2. Perform the opposite operations in the reverse order of the Order of Operations.

3. Combine all like terms on the same side of the equals sign before doing #1.

Example B

Given the equation 2b3

+ 6a4=8, findb whena=1 and a= 2.

Solution: This example combines what was learned in the last section with what we did in the previous example.

First, isolateb. Move both the 6aand the 4 over to the other side by doing the opposite operation.

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2b

3 + 6a4=8

6a + 4= +46a

2b

3 =126a

Now, we have a fraction multiplied by b. To undo this we must multiply by the reciprocal of 23

, which is 32

. This

means, we must multiply everything in the equation by the reciprocal.

2b

3 =126a

3

2

2b

3 =126a

3

2

2b

3 =

3

212

3

26a

b=189a

Even though we know that a 9 can be pulled out of the 18 and the 9 in the above equation, it is not a necessary step

in solving forb. Now, we can do what the question asks, findb whena=1 and -2. Plug in these values fora.

a=1 : b=189(1) =189=9 a= 2 : 189(2) =18 + 18=36

Example C

The area of a triangle isA= 12

bh, whereb is the base of the triangle and h is the height. You know that the area of a

triangle is 60in2 and the base is 12 in. Find the height.

Solution:First solve the equation forh.

A=1

2bh

2 A=2 1

2bh Multiply both sides by 2.

2A=bh Divide both side byb.

2A

b=h

Now, plug in what we know to findh. 2(60)

12 = 120

12 =10. The height is 10 inches.

This process is helpful when solving for any variable in any equation or formula. Some formulas you may need to

know for this text are listed below.

TABLE 1.1: Helpful Formulas

Distance d=rt d=distance,r=rate,t= timeTemperature F= 9

5C+ 32 F = degrees in Fahrenheit,C =

degrees in Celsius

Area of a Triangle A= 12

bh A=area,b=base,h=height

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TABLE 1.1: (continued)

Distance d=rt d=distance,r=rate,t= timeArea of a Rectangle A=bh A=area,b=base,h=heightArea of a Circle A= r2 A=area,r=radiusArea of a Trapezoid A= 1

2h(b1+ b2) A = area,h = height,b1 =

one base,b2=other basePerimeter of a Rectangle P=2l + 2w P = perimeter, l = length,w =

width

Circumference of a Circle C=2r C=circumference,r=radius

Guided Practice

1. Solve 34x2y=15 forx.

2. If the temperature is 41F, what is it in Celsius?

Answers

1. First add 2yto both sides, then multiply both sides by the reciprocal of 34

.

3

4x2y=15

+2y + 2y

3

4x=2y + 15

x=4

32y +

4

315

x=8

3y + 20

2. You can solve this problem two different ways: First, plug in 41 for Fand then solve for Cor solve for Cfirst and

then plug in 41 forF. We will do the second option.

F=9

5C+ 32

F32=9

5C

5(F32)

9 = C

Now, plug in 41 forF.

C=5(4132)

9

C=5 9

9

C=5

.

Problem Set

Solve the following equations or formulas for the indicated variable.

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1. 6x3y=9; solve fory.2. 4c + 9d= 16; solve forc.3. 5f6g=14; solve for f.4. 1

3x + 5y=1; solve for x.

5. 45

m + 23

n=24; solve for m.6. 4

5m + 2

3n=24; solve for n.

7. P=2l + 2w; solve forw.

8. F= 95C+ 32; solve forC.

Find the value ofy given the value ofx.

9. 4x8y=2;x= 110. 2y5x=12;x=1611. 3x + 1

2y= 5;x=7

12. 14x + 2

3y18=0;x= 24

For questions 13-15, use the formulas in the chart above to answer the following questions.

13. If the temperature is 86

F, what is it in Celsius?14. If the area of a circle is 36cm2, what is the radius?

15. The area of a trapezoid is 72 f t2, the height is 8 ft and b1is 6, what is the length of the other base?

For questions 16-17, use the equation for the surface area of a cylinder, SA=2r2 + 2rh, whereris the radius andhis the height.

16. Solve the equation forh.

17. Findh if the surface area is 120cm2 and the radius is 6 cm.

18. ChallengeThe formula for the volume of a sphere is V= 43r3. Solve the equation for r, the radius.

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Simplifying Algebraic Expressions and Equations of Equations and Inequalities s v6 Xzn s1

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