Solving One Step Equations using inverse operations. 2-2 2-1 Solving One Step Equations using inverse operations. Holt Algebra 1 2-2 Lesson Presentation

Objective Solve one-step equations in one variable by using addition or subtraction. Solve one-step equations in one variable by using multiplication or division.

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

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

Example 1A: 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 

Example 1B: Solving Equations by Using Addition Solve the equation. Check your answer. = z – 7 16 5 Since is subtracted from z, add to both sides to undo the subtraction. 7 16 + 7 16 = z 3 4 Check = z – 7 16 5 To check your solution, substitute for z in the original equation. 3 4 3 4 5 16 7 – 5 16 

Check It Out! Example 1a 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 

Check It Out! Example 1c Solve the equation. Check your answer. 16 = m – 9 + 9 + 9 Since 9 is subtracted from m, add 9 to both sides to undo the subtraction. 25 = m Check 16 = m – 9 To check your solution, substitute 25 for m in the original equation. 16 25 – 9 16 16 

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.

Example 3: Solving Equations by Adding the Opposite Solve – + p = – . 2 11 5 + 5 11 5 11 Since – is added to p, add to both sides. p = 3 11 Check + p = – 2 11 5 – To check your solution, substitute for p in the original equation. 3 11 2 5 11 – 3 + 2 11 – 

Check It Out! Example 3a Solve –2.3 + m = 7. Check your answer. –2.3 + m = 7 +2.3 + 2.3 Since –2.3 is added to m, add 2.3 to both sides. m = 9.3 Check –2.3 + m = 7 To check your solution, substitute 9.3 for m in the original equation. –2.3 + 9.3 7 7 7 

Since – is added to z, add to both sides. + 3 4 z = 2 Check It Out! Example 3b 3 4 5 4 Solve – + z = . Check your answer. – + z = 5 4 3 3 4 Since – is added to z, add to both sides. + 3 4 z = 2 Check + z = 5 4 3 – To check your solution, substitute 2 for z in the original equation. – 5 3 4 + 2 5 4 

decrease in population Example 4: Application Over 20 years, the population of a town decreased by 275 people to a population of 850. Write and solve an equation to find the original population. original population minus current population decrease in population is p – d = c p – d = c Write an equation to represent the relationship. p – 275 = 850 Since 275 is subtracted from p, add 275 to both sides to undo the subtraction. + 275 + 275 p =1125 The original population was 1125 people.

Check It Out! Example 4 A person's maximum heart rate is the highest rate, in beats per minute, that the person's heart should reach. One method to estimate maximum heart rate states that your age added to your maximum heart rate is 220. Using this method, write and solve an equation to find a person's age if the person's maximum heart rate is 185 beats per minute.

Check It Out! Example 4 Continued age added to 220 maximum heart rate is a + r = 220 a + r = 220 Write an equation to represent the relationship. a + 185 = 220 Substitute 185 for r. Since 185 is added to a, subtract 185 from both sides to undo the addition. – 185 – 185 a = 35 A person whose maximum heart rate is 185 beats per minute would be 35 years old.

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

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

Example 1A: 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 

Example 1B: 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 

Example 2B: 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 

Check It Out! Example 2a 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 

Check It Out! Example 2b Solve the equation. Check your answer. 0.5y = –10 Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication. y = –20 Check 0.5y = –10 0.5(–20) –10 To check your solution, substitute –20 for y in the original equation. –10 –10 

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.

Example 3A: 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 

Example 3B: 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 Check It Out! Example 3a 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 

Example 4: Application 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. 1 4 one-fourth times earnings equals college fund Write an equation to represent the relationship. Substitute 285 for c. Since m is divided by 4, multiply both sides by 4 to undo the division. m = $1140 Ciro earned $1140 mowing lawns.

Write an equation to represent the relationship. Check it Out! Example 4 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 Write an equation to represent the relationship. Substitute 45 for d. 15 = h The plane was flying at 15,000 ft when the descent began.

Homework Problems to try before you come to class. Page80 #22 to 48 even Page87-88 #22-42 even Don’t forget to bring your textbook to class.