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Warm UpEvaluate.1. 0.51 + (0.29)
Give the opposite of each number.
3. 8 4.
Evaluate each expression for a = 3 and b = 2.5. a + 5 6. 12 b Evaluate each expression.
7.
8.
9.
0.8
23
–8
14 8
23
1
23
1.8
Solve one-step equations in one variable by using addition or subtraction. Solve one-step equations in one variable by using multiplication or division.
Objective
Vocabularyequationsolution of an equation
An equation is a mathematical statement that two expressions are equal.
A solution of an equation is a value of the variable that makes the equation true.
To find solutions, isolate the variable. A variable is isolated when it appears by itself on one side of an equation, and not at all on the other side.
Inverse Operations
Operation Inverse Operation
Addition Subtraction
Subtraction Addition
Isolate a variable by using inverse operations which "undo" operations on the variable.
An equation is like a balanced scale. To keep the balance, perform the same operation on both sides.
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. Check your answer.
Solving Equations by Using Addition
Since 8 is subtracted from y, add 8 to both sides to undo the subtraction.
y – 8 = 24 + 8 + 8
y = 32
Check y – 8 = 24
32 – 8 2424 24
To check your solution, substitute 32 for y in the original equation.
Solve the equation. Check your answer.
Try This!
Since 3.2 is subtracted from n, add 3.2 to both sides to undo the subtraction.
n – 3.2 = 5.6
+ 3.2 + 3.2
n = 8.8
Check n – 3.2 = 5.6
8.8 – 3.2 5.65.6 5.6
To check your solution, substitute 8.8 for n in the original equation.
Solve the equation. Check your answer.
Try This!
Since 6 is subtracted from k, add 6 to both sides to undo the subtraction.
–6 = k – 6 + 6 + 6
0 = k
Check –6 = k – 6
–6 0 – 6–6 –6
To check your solution, substitute 0 for k in the original equation.
Solve the equation. Check your answer.
Solving Equations by Using Subtraction
Since 17 is added to m, subtract 17 from both sides to undo the addition.
m + 17 = 33– 17 –17
m = 16
Check m + 17 = 33
16 + 17 3333 33
To check your solution, substitute 16 for m in the original equation.
Solve the equation. Check your answer.
Solving Equations by Using Subtraction
Since 1.8 is added to t, subtract 1.8 from both sides to undo the addition.
4.2 = t + 1.8 –1.8 – 1.8
2.4 = t
Check 4.2 = t + 1.8
4.2 2.4 + 1.84.2 4.2
To check your solution, substitute 2.4 for t in the original equation.
Solve the equation. Check your answer.
Try This!
Since 5 is added to k, subtract 5 from both sides to undo the subtraction.
–5 = k + 5 – 5 – 5 –10 = k
Check –5 = k + 5
–5 –10 + 5–5 –5
To check your solution, substitute –10 for k in the original equation.
Solve the equation. Check your answer.
Try This!
Since 6 is added to t, subtract 6 from both sides to undo the addition.
6 + t = 14– 6 – 6
t = 8
Check 6 + t = 14
6 + 8 1414 14
To check your solution, substitute 8 for t in the original equation.
Remember that subtracting is the same as adding the opposite. When solving equations, you will sometimes find it easier to add an opposite to both sides instead of subtracting.
Solve the equation.
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.
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.
Try This!
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.
Try This!
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.
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.
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.
Try This!
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.
Try This!
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.
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.
Solving Equations That Contain Fractions
= z3
16
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.Try This!
– = 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–
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 QuizSolve each equation.
1. r – 4 = –8
2. m + 13 = 58
3. 0.75 = n + 0.6
4. –5 + c = 22
5.
6.8y = 4
7.126 = 9q
8.
–4
45
0.15
27
21
–14
40