308 CHAPTER 6 Algebra, Graphs, and Functions Solving for a Variable in a Formula or Equation Often in mathematics and science courses, you are given a formula or an equation expressed in terms of one variable and asked to express it in terms of a different variable. For example, you may be given the formula P i r2 = and asked to solve the formula for r. To do so, treat each of the variables, except the one you are solving for, as if it were a constant. Then solve for the variable desired, using the properties previously discussed. Examples 4 through 6 show how to do this task. When graphing equations in Section 6.6, you will sometimes have to solve the equation for the variable y, as is done in Example 4. Profile in Mathematics Sophie Germain Because she was a woman, Sophie Germain (1776–1831) was denied admission to the École Polytechnic, the French academy of mathematics and science. Not to be stopped, she obtained lecture notes from courses in which she had an interest, including one taught by Joseph-Louis Lagrange. Under the pen name M. LeBlanc, she submitted a paper on analysis to Lagrange, who was so impressed with the report that he wanted to meet the author and personally congratulate “him.” When he found out that the author was a woman, he became a great help and encouragement to her. Lagrange introduced Germain to many of the French scientists of the time. In 1801, Germain wrote to the great German mathematician Carl Friedrich Gauss to discuss Fermat’s equation, x y z . n n n + = He commended her for showing “the noblest courage, quite extraordinary talents and a superior genius.” Example 4 Solving for a Variable in an Equation Solve the equation x y 2 5 10 0 + − = for y. Solution We need to isolate the term containing the variable y. Begin by adding 10 and subtracting x2 from both sides of the equation. + − = + − + = + + = − + + = − + = − + = − + = − + = − + = − + x y x y x y x x y x y x y x y x y x y x 2 5 10 0 2 5 10 10 0 10 2 5 10 2 2 5 2 10 5 2 10 5 5 2 10 5 2 10 5 2 5 10 5 2 5 2 Addition property Subtraction property Division property 7 Now try Exercise 35 Example 3 Calculating Slope The slope, m, of a line, which will be discussed in Section 6.6, can be determined using the formula m y y x x , 2 1 2 1 = − − where x y ( , ) 1 1 and x y ( , ) 2 2 are the coordinates of two points on the line. Use this formula to determine the slope of a line containing the points (3, 2) − and ( 1, 6). − Solution Here x y x 3, 2, 1, 1 1 2 = = − = − and y 6. 2 = Substituting these values into the formula we have the following. m y y x x 6 ( 2) 1 3 8 4 2 2 1 2 1 = − − = − − − − = − = − Thus, the slope of the line containing the points − (3, 2) and ( 1,6) − is 2. − 7 Now try Exercise 21 Some formulas contain subscripts . Subscripts are numbers (or letters) placed below and to the right of variables. They are used to help clarify a formula. For example, if two different amounts are used in a problem, they may be symbolized as A and A , 0 or A1 and A .2 Subscripts are read using the word sub ; for example, A0 is read “ A sub zero” and A1 is read “ A sub one.” Our next example involves variables with subscripts.
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