EQUATIONS & PROBLEM SOLVING. Mr. Velazquez Honors Precalculus
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1 EQUATIONS & PROBLEM SOLVING Mr. Velazquez Honors Precalculus
2 LINEAR EQUATIONS IN ONE VARIABLE Any equation with only one unknown can be algebraically rewritten into this form. 3x = 9 3x 9 = 0 1 x 5 = 2x x 5 = 0 3 x + 1 = 6 6x + 9 = 0
3 MANIPULATION OF EQUATIONS IN 1 VAR
4 SOLVING EQUATIONS IN ONE VARIABLE Examples: Simplify and solve.
5 LINEAR EQUATIONS WITH FRACTIONS
6 LINEAR EQUATIONS WITH FRACTIONS Examples: Solve for x. x 1 3 2x Solve for x. x 3 x 1 x
7 RATIONAL EQUATIONS Remember that when rational expressions are involved, we must remove values of x that would make a denominator equal to zero. Then when we solve for x, check it against the previously removed values.
8 RATIONAL EQUATIONS Examples: Solve the Rational Equation x 3 x 2 Solve the Rational Equation. x x x 2
9 SOLVING FOR A VARIABLE Solving for a specific variable means isolating that variable on one side of the equation. Usually, this is used to alter an existing formula. Solving for P A = P + Prt A = P 1 + rt A 1 + rt = P
10 SOLVING FOR A VARIABLE Practice: Solve for the given variable Solve for l Solve for p 1 p + 1 q = 1 f
11 EQUATIONS INVOLVING ABSOLUTE VALUE Essentially, this means that an equation containing an absolute value will probably have two solutions. For example, consider the following equation: This can be written in two ways: 2x = 10 2x = 10 AND 2x = 10 2x + 2 = 10 2x 3 1 = 10 2x = 8 2x = 14 x = 4 x = 7
12 EQUATIONS INVOLVING ABSOLUTE VALUE Examples Solve: x Solve: 2 2x
13 QUADRATIC EQUATIONS 2x = 0 x 2 + 7x + 12 = 0 4x 2 x = 5
14 QUADRATIC EQUATIONS Examples: Solve using factoring and/or the Zero-Product Principle 2x 5 3x + 4 = 0 x 2 3x 4 = 0 2x 2 7x 4 = 0
15 USING THE SQUARE ROOT PROPERTY
16 USING THE SQUARE ROOT PROPERTY Examples: x 4 2 = 25 4x 2 7 = 0
17 COMPLETING THE SQUARE
18 COMPLETING THE SQUARE Start Add 1 2 b 2 Result Factored Form x 2 + 6x = 0 x 2 4x = 0 x 2 20x = = 9 x 2 + 6x + 9 = 9 x = 9 = 4 x 2 4x + 4 = 4 x 2 2 = 4 = 100 x 2 20x = 100 x 10 2 = 100
19 COMPLETING THE SQUARE
20 COMPLETING THE SQUARE Examples: Solve by completing the square x x 3 = 0 x 2 8x + 13 = 0
21 THE QUADRATIC FORMULA Solve the equation using the quadratic formula. 2 x 6x 3 2x 2 + 4x = 5
22 THE QUADRATIC FORMULA
23 THE QUADRATIC FORMULA Use the discriminant to find the number and types of solutions, but don't solve the equation. 2 a. x 5x b. x 3x 9 2 c. 2x 4x 9
24 PROBLEM SOLVING WITH EQUATIONS WARNING: Be careful of word problems that use words like exceeds or no less than. Always read these problems more than once to ensure you understand the quantities being asked for.
25 PROBLEM SOLVING WITH EQUATIONS Spice Drops candy calorie count exceeds Smarties candy calorie count by 70 calories per serving. If the sum of one serving of each candy equals 170 calories find the calorie count of each kind of candy. Step 1: Represent one of the quantities Step 2: Represent the other quantity. Step 3:Write an equation in x that models the conditions. Step 4: Solve the equation and answer the question. Step 5: check the proposed solution.
26 PROBLEM SOLVING WITH EQUATIONS The percentage of women in the labor force and the percentage of men in the labor force is illustrated in the graph at left. The decrease yearly of men in the labor force is ¼% and the increase in women in the labor force is ½%. If there are presently 70 million men and 60 million women in the labor force, when will the number of both sexes be equal?
27 PROBLEM SOLVING WITH EQUATIONS In 2002, the median annual income for people with an advanced college degree was $73,000. This is a 170% increase over the median income in 1982 of people with an advanced degree. What were people with an advanced college degree making in 1982?
28 PROBLEM SOLVING WITH EQUATIONS The size of a television set is always given by the length of the diagonal of its screen, along with its aspect ratio. If the aspect ratio is 16:9, and the height of the screen is 36 inches, what is the diagonal length of his TV?
29 PROBLEM SOLVING WITH EQUATIONS The rectangular painting in the figure measures 12 inches by 16 inches, as shown. It includes a frame of uniform width around the edges of the painting. If the outer perimeter of the frame is 72 inches, find the width of the frame.
30 CLASSWORK & HOMEWORK Classwork: 1. IN YOUR JOURNAL: Summarize what you learned today 2. ON A SEPARATE SHEET: Complete the first five multiples of 10 from Chapter P.8 (Pg. 111 #10, 20, 30, 40 and 50) Homework: Section P.7 Pg , #1-99 (odds only) Due 9/15
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