Transcript Chapter 2, Section 1

Using the Addition Property of Equality
Addition Property of Equality
If A, B, andCare 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.
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Sec 10.1 - 1
Using the Addition Property of Equality
Note
Equations can be thought of in
terms of a balance. Thus, adding
the same quantity to each side
does not affect the balance.
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Sec 10.1 - 2
Using the Addition Property of Equality
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
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(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
Sec 10.1 - 3
Using the Addition Property of Equality
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.
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Sec 10.1 - 4
Using the Addition Property of Equality
Example 2 Solve each equation.
Our goal is to get an equivalent equation of the form x = a number.
(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
–8 = a
Check: 4(–8) + 8 = 3(–8) ?
–24 = –24
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Sec 10.1 - 5
Simplifying and Using the Addition Property of Equality
Check:
5((2 · –36) –3) – (11(–36) + 1) =
Example 3
Solve.
5(2b – 3) – (11b + 1) = 20
10b – 15 – 11b – 1 = 20
–b – 16 = 20
5(–72 –3) – (–396 + 1) =
5(–75) – (–395) =
–375 + 395 = 20
–b – 16 + 16 = 20 + 16
–b = 36
b = –36
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Sec 10.1 - 6
Solving a Linear Equation
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.
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Sec 10.3 - 7
Using the Four Steps for Solving a Linear Equation
Example 1 Solve the equation.
Step 1
Step 2
Step 3
5w + 3 – 2w – 7 = 6w + 8
3w – 4 = 6w + 8
3w – 4 + 4 = 6w + 8 + 4
3w
3w – 6w
– 3w
– 3w
–3
w
= 6w + 12
= 6w + 12 – 6w
= 12
12
=
–3
= –4
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Combine terms.
Add 4.
Combine terms.
Subtract 6w.
Combine terms.
Divide by –3.
Sec 10.3 - 8
Using the Four Steps for Solving a Linear Equation
Example 1 (continued) Solve the equation.
Step 4
Check by substituting – 4 for w in the original equation.
5w + 3 – 2w – 7 = 6w + 8
5(– 4) + 3 – 2(– 4) – 7 = 6(– 4) + 8
– 20 + 3 + 8 – 7 = – 24 + 8
– 16 = – 16
?
Let w = – 4.
?
Multiply.
True
The solution to the equation is – 4.
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Sec 10.3 - 9
Using the Four Steps for Solving a Linear Equation
Example 2 Solve the equation.
Step 1
Step 2
Step 3
5(h – 4) + 2
5h – 20 + 2
5h – 18
5h – 18 + 18
5h
5h – 3h
2h
=
=
=
=
=
=
=
3h
3h
3h
3h
3h
3h
14
2h = 14
2
2
h = 7
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–
–
–
–
+
+
4
4
4
4 + 18
14
14 – 3h
Distribute.
Combine terms.
Add 18.
Combine terms.
Subtract 3h.
Combine terms.
Divide by 2.
Sec 10.3 - 10
Using the Four Steps for Solving a Linear Equation
Example 2 (continued) Solve the equation.
Step 4
Check by substituting 7 for h in the original equation.
5 ( h – 4 ) + 2 = 3h – 4
5 ( 7 – 4 ) + 2 = 3(7) – 4
?
Let h = 7.
5 (3) + 2 = 3(7) – 4
?
Subtract.
15 + 2 = 21 – 4
?
Multiply.
17 = 17
True
The solution to the equation is 7.
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Sec 10.3 - 11
Using the Four Steps for Solving a Linear Equation
Example 3 Solve the equation.
2 ( 5y + 7 ) – 16
10y + 14 – 16
10y – 2
Step 2
10y – 2 – 2
10y – 4
10y – 4 – 10y
–4
= –4
Step 3
–5
= 4
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Step 1
15y – 1 ( 10y – 2 )
15y – 10y + 2
5y + 2
5y + 2 – 2
5y
5y – 10y
–5y
–5y
–5
y
=
=
=
=
=
=
=
Distribute.
Combine terms.
Subtract 2.
Combine terms.
Subtract 10y.
Combine terms.
Divide by –5.
Sec 10.3 - 12
2.3 Applications of Linear Equations
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
Addition
The sum of a number and 2
x+2
3 more than a number
x+3
7 plus a number
7+x
16 added to a number
x + 16
A number increased by 9
x+9
The sum of two numbers
x+y
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Sec 2.3 - 13
2.3 Applications of Linear Equations
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
Subtraction
4 less than a number
x–4
10 minus a number
10 – x
A number decreased by 5
x–5
A number subtracted from 12
12 – x
The difference between two
numbers
x–y
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Sec 2.3 - 14
2.3 Applications of Linear Equations
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
Multiplication
14 times a number
14x
A number multiplied by 8
8x
3 of a number (used with
4
fractions and percent)
3 x
4
Triple (three times) a number
3x
The product of two numbers
xy
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Sec 2.3 - 15
2.3 Applications of Linear Equations
Translating from Words to Mathematical Expressions
Verbal Expression
Mathematical Expression
(where x and y are numbers)
Division
The quotient of 6 and a number
6 (x ≠ 0)
x
A number divided by 15
x
15
The ratio of two numbers
or the quotient of two numbers
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x (y ≠ 0)
y
Sec 2.3 - 16
2.3 Applications of Linear Equations
Caution
CAUTION
Because subtraction and division are not commutative operations, be careful
to correctly translate expressions involving them. For example, “5 less than a
number” is translated as x – 5, not 5 – x. “A number subtracted from 12” is
expressed as 12 – x, not x – 12.
For division, the number by which we are dividing is the denominator, and
the number into which we are dividing is the numerator. For example, “a
x
number divided by 15” and “15 divided into x” both translate as 15 . Similarly,
x
“the quotient of x and y” is translated as y .
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Sec 2.3 - 17
2.3 Applications of Linear Equations
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 =.
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Sec 2.3 - 18
2.3 Applications of Linear Equations
Translating Words into Equations
Verbal Sentence
Equation
Twice a number, decreased by 4, is 32.
2x – 4 = 32
If the product of a number and 16 is decreased
by 25, the result is 87.
16x – 25 = 87
The quotient of a number and the number plus
6 is 48.
The quotient of a number and 8, plus the
number, is 54.
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x = 48
x+6
x + x = 54
8
Sec 2.3 - 19