Warm Up 0.8 2 3 2 3 –8 8 14 1 2 3 1.8 Evaluate. 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 2 3 2 3 –8 8 14 1 2 3 1.8

Objective Vocabulary equation solution of an equation Solve one-step equations in one variable by using addition or subtraction. Solve one-step equations in one variable by using multiplication or division. Vocabulary equation solution 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 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 Addition Subtraction Subtraction Addition

Inverse Operations Operation Inverse Operation 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. Inverse Operations Operation Inverse Operation Multiplication Division Division Multiplication

Solving Equations by Using Addition Solve the equation. Check your answer. y – 8 = 24 + 8 + 8 Since 8 is subtracted from y, add 8 to both sides to undo the subtraction. y = 32 Check y – 8 = 24 To check your solution, substitute 32 for y in the original equation. 32 – 8 24 24 24 

Try This! Solve the equation. Check your answer. n – 3.2 = 5.6 + 3.2 + 3.2 Since 3.2 is subtracted from n, add 3.2 to both sides to undo the subtraction. n = 8.8 Check n – 3.2 = 5.6 To check your solution, substitute 8.8 for n in the original equation. 8.8 – 3.2 5.6 5.6 5.6 

Try This! Solve the equation. Check your answer. –6 = k – 6 + 6 + 6 Since 6 is subtracted from k, add 6 to both sides to undo the subtraction. 0 = k Check –6 = k – 6 To check your solution, substitute 0 for k in the original equation. –6 0 – 6 –6 –6 

Solving Equations by Using Subtraction Solve the equation. Check your answer. m + 17 = 33 – 17 –17 Since 17 is added to m, subtract 17 from both sides to undo the addition. m = 16 Check m + 17 = 33 To check your solution, substitute 16 for m in the original equation. 16 + 17 33 33 33 

Solving Equations by Using Subtraction Solve the equation. Check your answer. 4.2 = t + 1.8 –1.8 – 1.8 Since 1.8 is added to t, subtract 1.8 from both sides to undo the addition. 2.4 = t Check 4.2 = t + 1.8 To check your solution, substitute 2.4 for t in the original equation. 4.2 2.4 + 1.8 4.2 4.2 

Try This! Solve the equation. Check your answer. –5 = k + 5 – 5 – 5 Since 5 is added to k, subtract 5 from both sides to undo the subtraction. –10 = k Check –5 = k + 5 To check your solution, substitute –10 for k in the original equation. –5 –10 + 5 –5 –5 

Try This! Solve the equation. Check your answer. 6 + t = 14 – 6 – 6 Since 6 is added to t, subtract 6 from both sides to undo the addition. t = 8 Check 6 + t = 14 To check your solution, substitute 8 for t in the original equation. 6 + 8 14 14 14 

Remember that subtracting is the same as adding the opposite 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.

Solving Equations by Using Multiplication Solve the equation. –8 = j 3 Since j is divided by 3, multiply both sides by 3 to undo the division. –24 = j Check –8 = j 3 –8 –24 3 To check your solution, substitute –24 for j in the original equation. –8 –8 

Solving Equations by Using Multiplication Solve the equation. = 2.8 n 6 Since n is divided by 6, multiply both sides by 6 to undo the division. n = 16.8 Check = 2.8 n 6 2.8 16.8 6 To check your solution, substitute 16.8 for n in the original equation. 2.8 2.8 

Try This! Solve the equation. Check your answer. = 10 p 5 Since p is divided by 5, multiply both sides by 5 to undo the division. p = 50 Check = 10 p 5 10 50 5 To check your solution, substitute 50 for p in the original equation. 10 10 

Try This! Solve the equation. Check your answer. –13 = y 3 Since y is divided by 3, multiply both sides by 3 to undo the division. –39 = y y Check –13 = 3 –13 –39 3 To check your solution, substitute –39 for y in the original equation. –13 –13 

Solving Equations by Using Division Solve the equation. Check your answer. 9y = 108 Since y is multiplied by 9, divide both sides by 9 to undo the multiplication. y = 12 Check 9y = 108 To check your solution, substitute 12 for y in the original equation. 9(12) 108 108 108 

Solving Equations by Using Division Solve the equation. Check your answer. –4.8 = –6v Since v is multiplied by –6, divide both sides by –6 to undo the multiplication. 0.8 = v Check –4.8 = –6v –4.8 –6(0.8) To check your solution, substitute 0.8 for v in the original equation. –4.8 –4.8 

Try This! Solve the equation. Check your answer. 16 = 4c Since c is multiplied by 4, divide both sides by 4 to undo the multiplication. 4 = c Check 16 = 4c 16 4(4) To check your solution, substitute 4 for c in the original equation. 16 16 

Try This! Solve the equation. Check your answer. 15k = 75 Since k is multiplied by 15, divide both sides by 15 to undo the multiplication. k = 5 Check 15k = 75 To check your solution, substitute 5 for k in the original equation. 15(5) 75 75 75 

Remember that dividing is the same as multiplying by the reciprocal 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.

Solving Equations That Contain Fractions Solve the equation. 5 w = 20 6 The reciprocal of is . Since w is multiplied by , multiply both sides by . 5 6 w = 24 Check w = 20 5 6 To check your solution, substitute 24 for w in the original equation. 20 20 20 

Solving Equations That Contain Fractions Solve the equation. 3 1 8 = z 16 The reciprocal of is 8. Since z is multiplied by , multiply both sides by 8. 1 8 = z 3 2 Check 1 8 3 16 = z To check your solution, substitute for z in the original equation. 3 2 3 16 

Solve the equation. Check your answer. 1 4 1 5 – = b Try This! Solve the equation. Check your answer. 1 4 1 5 – = b The reciprocal of is 5. Since b is multiplied by , multiply both sides by 5. 1 5 = b 5 4 – = b 5 4 Check 1 – To check your solution, substitute – for b in the original equation. 5 4 

Properties of Equality 3 = 3 3 + 2 = 3 + 2 5 = 5 a = b a + c = b + c WORDS Addition Property of Equality You 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 7 = 7 7 – 5 = 7 – 5 2 = 2 a = b a – c = b – c WORDS Subtraction Property of Equality You 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 6 = 6 6(3) = 6(3) 18 = 18 a = b ac = bc WORDS Multiplication Property of Equality You 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 Division Property of Equality You 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 8 4 = a c

Lesson Quiz 1. r – 4 = –8 2. m + 13 = 58 –4 3. 0.75 = n + 0.6 Solve each equation. 1. r – 4 = –8 2. m + 13 = 58 3. 0.75 = n + 0.6 4. –5 + c = 22 5. 8y = 4 126 = 9q –4 45 0.15 27 21 –14 40