copyright © 2010 pearson education, inc. all rights reserved. sec 10.1 - 1 addition property of...

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 10.1 - 1

Addition Property of EqualityIf A, B, and C are real numbers, then the equations

A = B and A + C = B + C

are equivalent equations.

In words, we can add the same number to each side of an equation without changing the solution.

Using the Addition Property of Equality


NoteEquations can be thought of in terms of a balance. Thus, adding the same quantity to each side does not affect the balance.



Example 1 Solve each equation.

Our goal is to get an equivalent equation of the form x = a number.

(a) x – 23 = 8

x – 23 + 23 = 8 + 23


x = 31

Check: 31 – 23 = 8

(b) y – 2.7 = –4.1

y – 2.7 + 2.7 = –4.1 + 2.7

y = – 1.4

Check: –1.4 – 2.7 = –4.1



The same number may be subtracted from each side of an equation without changing the solution.

If a is a number and –x = a, then x = –a.


(a) –12 = z + 5

–12 – 5 = z + 5 – 5


–17 = z

Check: –12 = –17 + 5

(b) 4a + 8 = 3a

4a – 4a + 8 = 3a – 4a

8 = –a

Check: 4(–8) + 8 = 3(–8) ?

–8 = a

–24 = –24

Example 2 Solve each equation.

Our goal is to get an equivalent equation of the form x = a number.


Example 3

Solve.

5(2b – 3) – (11b + 1) = 20

10b – 15 – 11b – 1 = 20

Simplifying and Using the Addition Property of Equality

–b – 16 = 20

Check: 5((2 · –36) –3) – (11(–36) + 1) =

–b – 16 + 16 = 20 + 16

–b = 36

b = –36

5(–72 –3) – (–396 + 1) =

5(–75) – (–395) =

–375 + 395 = 20


Solving a Linear Equation

Step 1 Simplify each side separately. Clear (eliminate) parentheses, fractions, and decimals, using the distributive property as needed,

and combine like terms.

Step 2 Isolate the variable term on one side. Use the addition property so that the variable term is on one side of the equation and a number is on the other.

Step 3 Isolate the variable. Use the multiplication property to get the equation in the form x = a number, or a number = x. (Other letters may be used for the variable.)

Step 4 Check. Substitute the proposed solution into the original equation to see if a true statement results.

Solving a Linear Equation


Using the Four Steps for Solving a Linear Equation

5w + 3 – 2w – 7 = 6w + 8

3w – 4 = 6w + 8 Combine terms.

3w – 4 + 4 = 6w + 8 + 4

Step 1

Step 2 Add 4.

3w = 6w + 12 Combine terms.

3w – 6w = 6w + 12 – 6w Subtract 6w.

Combine terms.– 3w = 12

– 3 – 3Divide by –3.– 3w 12

=

w = – 4

Step 3

Example 1 Solve the equation.



5w + 3 – 2w – 7 = 6w + 8

? Let w = – 4.

Step 4

5(– 4) + 3 – 2(– 4) – 7 = 6(– 4) + 8

Check by substituting – 4 for w in the original equation.

– 20 + 3 + 8 – 7 = – 24 + 8 ? Multiply.

– 16 = – 16 True

The solution to the equation is – 4.

Example 1 (continued) Solve the equation.

2h 14



5 ( h – 4 ) + 2 = 3h – 4

5h – 20 + 2 = 3h – 4 Distribute.

5h – 18 = 3h – 4

Step 1

Step 2

Combine terms.

5h – 18 + 18 = 3h – 4 + 18 Add 18.

5h = 3h + 14 Combine terms.

Subtract 3h.5h – 3h = 3h + 14 – 3h

2 2

Combine terms.2h = 14

=

h = 7

Step 3 Divide by 2.




Check by substituting 7 for h in the original equation.Step 4

5 ( h – 4 ) + 2 = 3h – 4

5 ( 7 – 4 ) + 2 = 3(7) – 4

5 (3) + 2 = 3(7) – 4

15 + 2 = 21 – 4

17 = 17

? Let h = 7.

? Subtract.

True

? Multiply.

The solution to the equation is 7.

Example 2 (continued) Solve the equation.

–5y – 4



15y – ( 10y – 2 ) = 2 ( 5y + 7 ) – 16

15y – 10y + 2 = 10y + 14 – 16 Distribute.

5y + 2 = 10y – 2

Step 1

Step 2

Combine terms.

5y + 2 – 2 = 10y – 2 – 2 Subtract 2.

5y = 10y – 4 Combine terms.

Subtract 10y.5y – 10y = 10y – 4 – 10y

–5 –5

Combine terms.–5y = – 4

=

y =

Step 3 Divide by –5.

1

45



2.3 Applications of Linear Equations

Translating from Words to Mathematical Expressions

Verbal Expression

The sum of a number and 2

Mathematical Expression(where x and y are numbers)

Addition

3 more than a number

7 plus a number

16 added to a number

A number increased by 9

The sum of two numbers

x + 2

x + 3

7 + x

x + 16

x + 9

x + y




Verbal Expression

4 less than a number


Subtraction

10 minus a number

A number decreased by 5

A number subtracted from 12

The difference between two numbers

x – 4

10 – x

x – 5

12 – x

x – y




Verbal Expression

14 times a number


Multiplication

A number multiplied by 8

Triple (three times) a number

The product of two numbers

14x

8x

3x

xy

of a number (used withfractions and percent)

34 x3

4




Verbal Expression

The quotient of 6 and a number


Division

A number divided by 15

The ratio of two numbersor the quotient of two numbers

(x ≠ 0)6x

(y ≠ 0)xy

x15


CAUTION

Because subtraction and division are not commutative operations, be carefulto correctly translate expressions involving them. For example, “5 less than anumber” is translated as x – 5, not 5 – x. “A number subtracted from 12” isexpressed as 12 – x, not x – 12. For division, the number by which we are dividing is the denominator, andthe number into which we are dividing is the numerator. For example, “a number divided by 15” and “15 divided into x” both translate as . Similarly,“the quotient of x and y” is translated as .


Caution

x15x

y



Indicator Words for Equality

Equality

The symbol for equality, =, is often indicated by the word is. In fact, any words that indicate the idea of “sameness” translate to =.



Translating Words into Equations

Verbal Sentence Equation

16x – 25 = 87If the product of a number and 16 is decreasedby 25, the result is 87.

= 48The quotient of a number and the number plus 6 is 48.

x + 6x

+ x = 54The quotient of a number and 8, plus the number, is 54.

8x

Twice a number, decreased by 4, is 32. 2x – 4 = 32

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