Algebra unit 2.1.1
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Transcript of Algebra unit 2.1.1
UNIT 2.1 ONE STEP EQUATIONS WITH UNIT 2.1 ONE STEP EQUATIONS WITH
MULTIPLICATION AND DIVISIONMULTIPLICATION AND DIVISION
Warm UpEvaluate each expression. 1. (–7)(2.8)
2. 0.96 ÷ 6
3. (–9)(–9)
4.
5.
6.
−19.6
0.16
1
81
23
1.8
Inverse Operations
Operation Inverse Operation
Multiplication Division
Division Multiplication
Solving an equation that contains multiplication or division is similar to solving an equation that contains addition or subtraction. Use inverse operations to undo the operations on the variable.
Solve the equation.
Example 1A: Solving Equations by Using Multiplication
Since j is divided by 3, multiply both sides by 3 to undo the division.–24 = j
–8 –8
To check your solution, substitute –24 for j in the original equation.
–8 = j3
–8 –243
Check –8 =j3
Solve the equation.
Example 1B: Solving Equations by Using Multiplication
Since n is divided by 6, multiply both sides by 6 to undo the division.n = 16.8
2.8 2.8
To check your solution, substitute 16.8 for n in the original equation.
= 2.8n6
2.8 16.86
Check = 2.8n6
Solve the equation. Check your answer.
Check It Out! Example 1a
Since p is divided by 5, multiply both sides by 5 to undo the division.p = 50
10 10
To check your solution, substitute 50 for p in the original equation.
= 10p5
10 505
Check = 10p5
Solve the equation. Check your answer.
Check It Out! Example 1b
Since y is divided by 3, multiply both sides by 3 to undo the division.–39 = y
–13 –13
To check your solution, substitute –39 for y in the original equation.
–13 = y3
–13 –393
yCheck –13 = 3
Solve the equation. Check your answer.
Check It Out! Example 1c
Since c is divided by 8, multiply both sides by 8 to undo the division.c = 56
7 7
To check your solution, substitute 56 for c in the original equation.
= 7c8
7 568
Check = 7c8
Solve the equation. Check your answer.
Example 2A: Solving Equations by Using Division
Since y is multiplied by 9, divide both sides by 9 to undo the multiplication.y = 12
108 108
To check your solution, substitute 12 for y in the original equation.
9y = 108
9(12) 108
Check 9y = 108
Solve the equation. Check your answer.
Example 2B: Solving Equations by Using Division
Since v is multiplied by –6, divide both sides by –6 to undo the multiplication.
0.8 = v
–4.8 –4.8
To check your solution, substitute 0.8 for v in the original equation.
–4.8 = –6v
–4.8 –6(0.8)
Check –4.8 = –6v
Solve the equation. Check your answer.
Check It Out! Example 2a
Since c is multiplied by 4, divide both sides by 4 to undo the multiplication.
4 = c
16 16
To check your solution, substitute 4 for c in the original equation.
16 = 4c
16 4(4)
Check 16 = 4c
Solve the equation. Check your answer.
Check It Out! Example 2b
Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication.y = –20
–10 –10
To check your solution, substitute –20 for y in the original equation.
0.5y = –10
0.5(–20) –10
Check 0.5y = –10
Solve the equation. Check your answer.
Check It Out! Example 2c
Since k is multiplied by 15, divide both sides by 15 to undo the multiplication.k = 5
75 75
To check your solution, substitute 5 for k in the original equation.
15k = 75
15(5) 75
Check 15k = 75
Remember that dividing is the same as multiplying by the reciprocal. When solving equations, you will sometimes find it easier to multiply by a reciprocal instead of dividing. This is often true when an equation contains fractions.
Solve the equation.
Example 3A: Solving Equations That Contain Fractions
w = −24
−20 −20
To check your solution, substitute −24 for w in the original equation.
w = −2056
Check w = −2056
The reciprocal of is . Since w is
multiplied by , multiply both sides
by .
56
65
566
5
−20
Solve the equation.
Example 3B: Solving Equations That Contain Fractions
= z316
To check your solution,
substitute for z in the
original equation.
32
= z32
18
The reciprocal of is 8. Since z is
multiplied by , multiply both sides
by 8.
181
8
Check18
316 = z
316
316
Solve the equation. Check your answer.Check It Out! Example 3a
– = b14
To check your solution,
substitute – for b in the
original equation.
54
15
The reciprocal of is 5. Since b is
multiplied by , multiply both sides
by 5.
151
5
= b54–
= b5 4Check11–
Check It Out! Example 3b
j = 1
Solve the equation.
=4j6
23
is the same as j.46
4j6
The reciprocal of is . Since j is
multiplied by , multiply both sides
by .
464
6
64
64
Check It Out! Example 3b Continued
To check your solution,
substitute 1 for j in the
original equation.
Check23
4j6 =
Solve the equation.
=4j6
23
w = 712
102 102
To check your solution, substitute 712 for w in the original equation.
w = 10216
Check w = 10216
The reciprocal of is 6. Since w is
multiplied by , multiply both sides
by 6 .
16
16
Solve the equation.Check It Out! Example 3c
102
Example 4: Application
Write an equation to represent the relationship.
Ciro puts of the money he earns from mowing lawns into a college education fund. This year Ciro added $285 to his college education fund. Write and solve an equation to find how much money Ciro earned mowing lawns this year.
14
one-fourth times earnings equals college fund
m = $1140
Substitute 285 for c. Since m is divided by 4, multiply both sides by 4 to undo the division.
Ciro earned $1140 mowing lawns.
Check it Out! Example 4
Write an equation to represent the relationship.
The distance in miles from the airport that a plane should begin descending, divided by 3, equals the plane's height above the ground in thousands of feet. A plane began descending 45 miles from the airport. Use the equation to find how high the plane was flying when the descent began.
Distance divided by 3 equals height in thousands of feet
15 = h
Substitute 45 for d.
The plane was flying at 15,000 ft when the descent began.
WORDS
Addition Property of EqualityYou can add the same number to both sides of an equation, and the statement will still be true.
NUMBERS
3 = 3 3 + 2 = 3 + 2 5 = 5
ALGEBRA a = b a + c = b + c
Properties of Equality
WORDS
Subtraction Property of EqualityYou can subtract the same number from both sides of an equation, and the statement will still be true.
NUMBERS
7 = 7 7 – 5 = 7 – 5 2 = 2
ALGEBRA a = b a – c = b – c
Properties of Equality
WORDS
Multiplication Property of EqualityYou can multiply both sides of an equation by the same number, and the statement will still be true.
NUMBERS
6 = 6 6(3) = 6(3) 18 = 18
ALGEBRA a = b ac = bc
Properties of Equality
Properties of EqualityDivision Property of EqualityYou can divide both sides of an equation by the same nonzero number, and the statement will still be true.
WORDS
a = b (c ≠ 0)
8 = 8
2 = 2
ALGEBRA
NUMBERS 84
84
=
ac
ac=
Lesson Quiz: Part 2
7. A person's weight on Venus is about his or her weight on Earth. Write and solve an equation to find how much a person weighs on Earth if he or she weighs 108 pounds on Venus.
910
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