Here are the steps required for Solving Combined Variation Problems:

 Step 1: Draw a right triangle and then read through the problems again to determine the length of the legs and the hypotenuse. Step 2: Use the Pythagorean Theorem (a2 + b2 = c2) to write an equation to be solved. Remember that a and b are the legs and c is the hypotenuse (the longest side or the side opposite the 90º angle). Step 3: Simplify the equation by distributing and combining like terms as needed. Step 4: Solve the equation. If the equation contains x2, set the equation equal to zero and solve the equation by factoring. If the equation does not contain x2, then solve the equation by getting the variables on one side and the numbers on the other side. Step 5: Answer the question asked in the original question and make sure that the answer makes sense. Do not forget to include units in your final answer.

Example 1 – The hypotenuse of a right triangle is 1 inch longer than the longer leg. The shorter leg is 7 inches shorter than the longer leg. Find the length of the hypotenuse.

 Step 1: Draw a right triangle and then read through the problems again to determine the length of the legs and the hypotenuse. Step 2: Use the Pythagorean Theorem (a2 + b2 = c2) to write an equation to be solved. Step 3: Simplify the equation by distributing and combining like terms as needed. Step 4: Solve the equation. In the case, we need to get the equation equal to zero and solve by factoring. Step 5: Answer the question asked in the original question and make sure that the answer makes sense. Do not forget to include units in your final answer. In this case, x = 4 does not make sense because 4 – 7 = –3 which is impossible, so x = 12 is the correct answer. Example 2 –Carrie works dues south of her apartment. Her friend Sarah works due east of the apartment. They leave for work at the same time. By the time Carrie is 5 miles form their apartment, the distance between them is 1 mile more than Sarah’s distance from the apartment. How far from the apartment is Sarah?

 Step 1: Draw a right triangle and then read through the problems again to determine the length of the legs and the hypotenuse. Step 2: Use the Pythagorean Theorem (a2 + b2 = c2) to write an equation to be solved. Step 3: Simplify the equation by distributing and combining like terms as needed. Step 4: Solve the equation. In the case, the x2 will cancel out so we need to solve by getting the x’s on one side and the numbers on the other side. Step 5: Answer the question asked in the original question and make sure that the answer makes sense. Do not forget to include units in your final answer. Example 3 – A guy wire is attached to a telephone pole. The distance from the point where the wire touches the ground to the base of the telephone pole is 4 feet less than the length of the wire. How far up the telephone pole is the wire attached if the distance from the ground to where the wire is attached to the pole is 2 feet less than the length of the wire?

 Step 1: Draw a right triangle and then read through the problems again to determine the length of the legs and the hypotenuse. Step 2: Use the Pythagorean Theorem (a2 + b2 = c2) to write an equation to be solved. Step 3: Simplify the equation by distributing and combining like terms as needed. Step 4: Solve the equation. In the case, we need to get the equation equal to zero and solve by factoring. Step 5: Answer the question asked in the original question and make sure that the answer makes sense. Do not forget to include units in your final answer. In this case, x = 2 does not make sense because 2 – 4 = –2 which is impossible, so x = 10 is the correct answer. Example 4 – The foot of a ladder is 10 feet from the wall. The ladder is 2 feet longer than the height that it reaches on the wall. What is the length of the ladder?

 Step 1: Draw a right triangle and then read through the problems again to determine the length of the legs and the hypotenuse. Step 2: Use the Pythagorean Theorem (a2 + b2 = c2) to write an equation to be solved. Step 3: Simplify the equation by distributing and combining like terms as needed. Step 4: Solve the equation. In the case, the x2 will cancel out so we need to solve by getting the x’s on one side and the numbers on the other side. Step 5: Answer the question asked in the original question and make sure that the answer makes sense. Do not forget to include units in your final answer. 