8-1 Variation Functions The variable y varies directly as the variable if y k for some constant k. To solve direct variation problems: k is called the constant of variation. Use the known and y values in the problem to solve for k. Write the direct variation equation, substituting the value for k. Use the direct variation equation to solve for the missing variable. If y varies directly as, and y 52 when 4, find y when 6. Step 1 Use y 52 when 4. y k 52 k 4 13 k Step 2 Write the direct variation equation. y k y 13 Step 3 Solve for y when 6. y 13 y 13 6 y 78 The variable y varies jointly as the variables and z if y kz for some constant k. Joint variation problems are solved like direct variation problems. If y varies jointly as and z, and y 90 when 36 and z 5, find y when 40 and z 3. Step 1 y kz 90 k 36 5 90 180k 0.5 k Step 2 Write the joint variation equation. y kz y 0.5z Step 3 Solve for y when 40 and z 3. y 0.5z y 0.5 40 3 y 60 Solve each problem. 1. If y varies directly as, and y 30 when 20, find y when 50. a. Step 1: b. Step 2: c. Step 3: y k y 1.5 y 1.5 30 k 20 y 1.5 50 k 1.5 y 75 2. If y varies jointly as and z, and y 150 when 2.5 and z 12, find y when 4 and z 6.5. a. Step 1: b. Step 2: c. Step 3: y kz ; 150 k 2.5 12; 150 30k, k 5 y 5z y 5z ; y 5 4 6.5; y 130 6 Holt Algebra 2
8-1 Variation Functions (continued) The variable y varies inversely as the variable if y k for some constant k. If y varies inversely as, and y 4 when 30, find y when 20. Step 1 Use y 4 when 30. Step 2 Write the inverse variation equation. Step 3 Solve for y when 20. y k 4 k y k y 120 30 120 k y 120 y 120 20 y 6 To graph the inverse variation function y 120, make a table of values. y y 10 12 10 12 20 6 20 6 30 4 30 4 40 3 40 3 Because the function is undefined for 0, make separate tables for negative and positive -values. Solve each problem. 3. If y varies inversely as, and y 2 when 9, find y when 6. Then graph the inverse variation function. a. Step 1: y k ; 2 k ; k 18 9 b. Step 2: y 18 c. Step 3: y 18 ; y 18 6 ; y 3 7 Holt Algebra 2
Solving Rational Equations and Inequalities To solve a rational equation, clear any denominators by multiplying each term on both sides of the equation by the least common denominator, LCD. Solve: 12 7. Step 1 The LCD is. Multiply each term by. 12 7 2 12 7 2 7 12 0 3 4 0 This makes the equation a quadratic equation. Set one side equal to 0 to solve a quadratic equation. Step 5 Set each factor equal to 0. 3 0 4 0 Step 6 3 4 Check 12 7 3 4 3 12 3 3 4 7Z 4 12 4 3 7Z 4 1. 2 1 4 2. 6 1 3. 4 5 2 2 1 2 4 2 6 1 Always check the solutions to rational equations. 4 5 2 2 8 2 6 2 4 5 2 2 8 0 2 6 0 2 4 5 0 4 2 0 3 2 0 5 1 0 4, 2 3, 2 5, 1 38 Holt Algebra 2
Solving Rational Equations and Inequalities (continued) Check all solutions to rational equations. If the solution to a rational equation makes the denominator equal to zero, then that solution is NOT a solution. It is called an etraneous solution. Solve: 4 6 2 10 6. Step 1 The LCD is 2 6. Multiply each term by 2 6. 4 6 2 6 2 2 6 10 6 2 6 2 4 6 10 2 2 8 2 6 20 Remember to multiply EVERY term by the LCD. 2 4 12 0 2 6 0 Step 5 Set each factor equal to 0 and solve. 2 0 6 0 2 6 Step 6 Check: 4 6 2 10 6 2 2 4 2 2 6 2 10 2 6? 2 8 1 10 8 Z The only solution is 2. 6 is etraneous. This value makes the denominators of the original equation equal to 0. 4. 1 1 2 2 5 5. 3 3 1 4 1 1 2 5 1 2 2 5 2 3 3 1 3 1 3 1 5 5 2 4 1 3 1 5 5 1 2 1 3 3 12 2 3 10 0; 5 2 2 3 0; 3 39 Holt Algebra 2
Solving Rational Equations and Inequalities To solve a rational equation, clear any denominators by multiplying each term on both sides of the equation by the least common denominator, LCD. Solve: 12 7. Step 1 The LCD is. Multiply each term by. 12 7 2 12 7 2 7 12 0 3 4 0 This makes the equation a quadratic equation. Set one side equal to 0 to solve a quadratic equation. Step 5 Set each factor equal to 0. 3 0 4 0 Step 6 3 4 Check 12 7 3 4 3 12 3 3 4 7Z 4 12 4 3 7Z 4 1. 2 1 4 2. 6 1 3. 4 5 2 2 1 2 4 2 6 1 Always check the solutions to rational equations. 4 5 2 2 8 2 6 2 4 5 2 2 8 0 2 6 0 2 4 5 0 4 2 0 3 2 0 5 1 0 4, 2 3, 2 5, 1 38 Holt Algebra 2
Solving Rational Equations and Inequalities (continued) Check all solutions to rational equations. If the solution to a rational equation makes the denominator equal to zero, then that solution is NOT a solution. It is called an etraneous solution. Solve: 4 6 2 10 6. Step 1 The LCD is 2 6. Multiply each term by 2 6. 4 6 2 6 2 2 6 10 6 2 6 2 4 6 10 2 2 8 2 6 20 Remember to multiply EVERY term by the LCD. 2 4 12 0 2 6 0 Step 5 Set each factor equal to 0 and solve. 2 0 6 0 2 6 Step 6 Check: 4 6 2 10 6 2 2 4 2 2 6 2 10 2 6? 2 8 1 10 8 Z The only solution is 2. 6 is etraneous. This value makes the denominators of the original equation equal to 0. 4. 1 1 2 2 5 5. 3 3 1 4 1 1 2 5 1 2 2 5 2 3 3 1 3 1 3 1 5 5 2 4 1 3 1 5 5 1 2 1 3 3 12 2 3 10 0; 5 2 2 3 0; 3 39 Holt Algebra 2
Reading Strategy Analyze Information The solutions to a rational function are those values that result from solving the equation. Some solutions, however, are etraneous. Etraneous solutions are solutions of a derived equation that are not solutions of the original equation. Find the solution. Write the equation. 2 2 4 4 4 4 4 4 4 4 2 4 4 Substitute the solution into the original equation. 2 4 4 4 4 4 4 4 Analyze. The solution gives a denominator of 0, so the solution is etraneous. This equation has no solution. Identify values of that would be etraneous solutions for each equation. 1. 6 2. 2 2 2 3 8 3 3. 6 3 1 4. 1 2 1 1 7. 2 0 3 1, 0, 1 5, 0 4 2 9 1 3 3, 3 2 3 5. 5 1 5 4 3 6. 7 2 3 2 2 8. 4 4 1 7 9. 2 3 2 2 2 8 0, 1 4 2, 1 Solve. 10. Ralph solved the inequality 2 1 1. He found the solutions to be 1 and 1 2. He knows the solution has to be epressed as an inequality. He thinks the solution should be written 1 or 1. Is he correct? How do you know? 2 No; possible answer: the solution should be 1 or 1 2. 1 is an etraneous solution. 2 42 Holt Algebra 2