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Unformatted text preview: have 63x + 45y = 135 25x – 45y = 85 88x = 220 x=
220 1 =2 88 2 1 for x in either equation and solve the resulting equation for y. 2 Since we were only asked to solve for x, we stop here. If we were asked to solve for both x and y, we would now substitute 2 7(2.5) + 5y = 15 17.5 + 5y = 15 5y = –2.5 1 y = –.5 or –
2 Example: Solve for x: ax + by = r cx – dy = s Solution: Multiply the first equation by d and the second by b to eliminate the y terms by addition. adx + bdy = dr bcx – bdy = bs adx + bcx = dr + bs Factor out x to determine the coefficient of x. x(ad + bc) = dr + bs x=
dr + bs ad + bc www.petersons.com 122 Chapter 8 Exercise 3
Work out each problem. Circle the letter that appears before your answer. 1. Solve for x: x – 3y = 3 2x + 9y = 11 (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 Solve for x: .6x + .2y = 2.2 .5x – .2y = 1.1 (A) 1 (B) 3 (C) 30 (D) 10 (E) 11 Solve for y: (A) (B) (C) (D) (E) 7b 2b 3b 1
2 7 1 7 4. If 2x = 3y and 5x + y = 34, find y. (A) 4 (B) 5 (C) 6 (D) 6.5 (E) 10 If x + y = –1 and x – y = 3, find y. (A) 1 (B) –2 (C) –1 (D) 2 (E) 0 5. 2. 3. 2x + 3y = 12b 3x – y = 7b –b www.petersons.com Concepts of Algebra—Signed Numbers and Equations 123 4. QUADRATIC EQUATIONS
In solving quadratic equations, there will always be two roots, even though these roots may be equal. A complete quadratic equation is of the form ax2 + bx + c = 0, where a, b, and c are integers. At the level of this examination, ax2 + bx + c can always be factored. If b and/or c is equal to 0, we have an incomplete quadratic equation, which can still be solved by factoring and will still have two roots. Example: x2 + 5x = 0 Solution: Factor out a common factor of x. x(x + 5) = 0 If the product of two factors is 0, either factor may be set equal to 0, giving x = 0 or x + 5 = 0. From these two linear equations, we find the two roots of the given quadratic equation to be x = 0 and x = –5. Example: 6x2 – 8x = 0 Solution: Factor out a common factor of 2x. 2x(3x – 4) = 0 Set each factor equal to 0 and solve the resulting linear equations...
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This note was uploaded on 08/15/2010 for the course MATH a4d4 taught by Professor Colon during the Spring '10 term at Embry-Riddle FL/AZ.
- Spring '10