WARM UP 54 2. Evaluate 3x + 5y + z for x = 18 y = 9 z= 4 103 1. Evaluate 3x + y for x = 16 y = 6 54 2. Evaluate 3x + 5y + z for x = 18 y = 9 z= 4 103 3. Use the Subtraction Theorem to write equivalent equation for -18x – 5y. -18x + (-5y) 4. Factor: a. 9p- 9 b. xy – x c.4a + 8b - 4 9(p - 1) x(y - 1) 4(a + 2b – 1)

SOLVING & WRITING EQUATIONS

Will the scale be balanced? WHAT IS AN EQUATION? A scale has several weights on one side and an object on the other. The scale is balanced. Will the scale be balanced?

AN EQUATION IS….. A mathematical sentence A = B says that the symbols A and B are equivalent. Such a sentence is an equation. The set of all acceptable replacements is the replacement set. The replacements that make an equation true are its solutions. The set of all solutions is the solution set. Unless otherwise stated, the replacement set we will use for solving equations and inequalities is the set of real numbers.

MORE ON EQUATIONS When we have found all the solutions of an equation, we say that we have solved the equation One approach to solving an equation is to transform it to a simpler equation whose solution set is obvious. The addition and multiplication properties of equality can be used when transforming equations.

PROPERTIES OF EQUALITY The Addition Property of Equality: If a = b, then a + c = b + c for any real number. Using the addition property, we can assume that the scale will balance. Suppose we triple the weight on both sides, will the scale balance? The Multiplication Property of Equality: If a = b, then a x c = b x c for any real number. Using the property of multiplication, we can assume the scale will balance.

EXAMPLE Solve 3x – 4 = 13 3x – 4 + 4 = 13 + 4 3x = 17 We often need to use the addition and multiplication properties together. We usually use the addition property first. Solve 3x – 4 = 13 Use the Addition Property add the inverse of -4 3x – 4 + 4 = 13 + 4 3x = 17 Use the multiplication property, multiply by the reciprocal of 3

TRY THIS…….. a. 13 = -25 + y b. -4x = 64 c. 9x – 4 = 8

WRITING EQUATIONS In your studies of mathematics, you have had considerable experience solving problems. The following guideline can help to solve many algebra problems. Phase 1: UNDERSTAND the problem: What am I trying to find? What data am I given? Have I ever solved a similar one. Phase 2: Develop and carry out a PLAN: What strategies might I use to solve the problem? How can I correctly carry out the strategies? Phase 3: Find the ANSWER and CHECK: Does the proposed solution check? What is the answer to the problem? Does the answer seem reasonable? Have I stated the answer clearly?

EXAMPLE The time that a traffic light remains yellow is 1 second more than 0.05 times the speed limit. What is the yellow time for a traffic light on a street with a speed limit of 30 mph. Question:What is the time that the traffic light remains yellow? Data: The yellow time is 0.05 times the speed limit plus 1 second. (Identifying the data) Develop and carry out a plan – choose a strategy: Write an equation:

EXAMPLE Yellow Time is 1 second more than 0.05 times the speed limit y = 1 + 0.05 x s y = 1 + 0.05s Translating into an equation For a speed limit of 30 mph, s will be 30. Thus we have the following: y = 1 + 0.05(30) Find the answer & check: On a 30 mph street, 2.5 seconds is a reasonable time for the light to remain yellow. y = 1 + 1.5 y = 2.5

TRY THIS…… a. The Country Cab Company charges 70¢ plus 12¢ per ¼ kilometer for each ride. What will be the total cost of a 14-km ride?

TRY THIS…… b. An insecticide originally contained ½ ounce of pyrethrums. The new formula contains ⅝ oz. of pyrethrums. What percent of the pyrethrums of the original formula does the new formula contain?