4. ; $22, $11, $10 5. C6. F7. B

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. Vocabulary

Inverse Operations OperationInverse Operation AdditionSubtraction 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.

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

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. Example 4: Application Substitute 275 for d and 850 for c p =1125 p – d = c original population minus current population decrease in population is p – 275 = 850 Since 275 is subtracted from p, add 275 to both sides to undo the subtraction. p – 275 =850 The original population was 1125 people.

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

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

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 = = = 5 ALGEBRA a = b a + c = b + c Properties of Equality

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

Inverse Operations OperationInverse Operation MultiplicationDivision 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 3A: Solving Equations That Contain Fractions w = 24 20 To check your solution, substitute 24 for w in the original equation. w =  Check w =  The reciprocal of is. Since w is multiplied by, multiply both sides by 20

Solve the equation. Example 3B: Solving Equations That Contain Fractions = z 3 16 = z The reciprocal of is 8. Since z is multiplied by, multiply both sides by

Solve the equation. Check your answer. Check It Out! Example 3a – = b The reciprocal of is 5. Since b is multiplied by, multiply both sides by = b 5 4 –

Check It Out! Example 3b j = 1 Solve the equation. = 4j4j is the same as j j 6 The reciprocal of is. Since j is multiplied by, multiply both sides by

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. 1 4 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 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 = a c a c =

Homework Assignment Sec. 1-2 & 1-3 Wkshts