6.7 Solving Systems of Linear Equations 361 Let = = x y number of m of 5% solution number of m of 25% solution We know that 500 m of solution are needed. Thus, x y 500 + = The total amount of phenobarbital in a solution is determined by multiplying the percent of phenobarbital by the number of milliliters of solution. The second equation comes from the fact that x y x y Total amount of phenobarbital in 5% solution total amount of phenobarbital in 25% solution total amount of phenobarbital in 10% mixture 0.05 0.25 0.10(500) or 0.05 0.25 50 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ + ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ = ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ + = + = The system of equations is + = + = x y x y 500 0.05 0.25 50 Let’s solve this system of equations by using the addition method. To obtain integer values in the second equation, we can multiply both sides of the equation by 100. The result will be an x-term of x5 . If we multiply both sides of the first equation by 5, − that will result in an x-term of x5 . − By following this process, we can eliminate the x-terms from the system. x y x y x y x y 5[ 500] gives 5 5 2500 100[0.05 0.25 50] gives 5 25 5000 − + = − − = − + = + = − − = − + = = = = x y x y y y y 5 5 2500 5 25 5000 20 2500 20 20 2500 20 125 Now we determine x. + = + = = x y x x 500 125 500 375 Therefore, 375 m of a 5% phenobarbital solution must be mixed with 125 m of a 25% phenobarbital solution to obtain 500 m of a 10% phenobarbital solution. 7 Now try Exercise 57 Example 10 can also be solved by using substitution. Exercises Warm Up Exercises In Exercises 1–10, fill in the blank with an appropriate word, phrase, or symbol(s). 1. When two or more linear equations are considered simultaneously, the equations are called a(n) ________ of linear equations. System 2. The ordered pair or ordered pairs that satisfy all equations in a system of equations is called the ________ to the system of equations. Solution 3. A system of equations that has no solution is called a(n) ________ system of equations. Inconsistent SECTION 6.7 Instructor Resources for Section 6.7 in MyLab Math • Objective-Level Videos 6.7 • Animation: Solving a Linear System by Graphing • PowerPoint Lecture Slides 6.7 • MyLab Exercises and Assignments 6.7
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