We Add The Resulting Equations To Get
2x 5y 19 2x 5y 19 0 0 As in Example 5, both x and y drop out. However, this time a true result is obtained. This shows that the equations are dependent and the system has infinitely many so- lutions. Any ordered pair that satisfies one equation satisfies the other also. Some so- lutions are , , and .
3x y 1 Self Check Solve the system: ⢠6 3 . â 0.3x 0.1y 0.2
To Find Y We Substitute 20 For X In Equation 6 And Solve For Y:
State the conclusion The store sold 20 of the less expensive radios and 16 of the more expensive radios.
Check the result If 20 of one type were sold and 16 of the other type were sold, a total of 36 radios were sold. Since the value of the less expensive radios is 20 $1,340 and the value of the more expensive radios is 16 $1,600, the total receipts are $2,940. â
We now review the graphing method of solving systems of two linear inequalities in two variables. Recall that the solutions are usually the intersection of half-planes.
x y 1 EXAM PLE 9 Graph the solution set of the system: e . 2x y 2
Solution On the same set of coordinates axes, we graph each inequality, as in Figure 13-7. 830 Chapter 13 More on Systems of Equations and Inequalities
The graph of the inequality x y 1 includes the line graph of the equation x y 1 and all points below it. Since the boundary line is included, it is drawn as a solid line. The graph of the inequality 2x y 2 contains only those points below the graph of the equation 2x y 2. Since the boundary line is not included, it is drawn as a broken line. The area where the half-planes intersect represents the simultaneous solution of the given system of inequalities, because any point in that region has coordinates that will satisfy both inequalities.
x+y=1 2x â y = 2
x y 1 2x y 2 x y x y 2x â y > 2 x 0 1 0 2 1 0 1 0 x+yâ¤1
Figure 13-7
Getting Ready Find Each Power
In this section, we will discuss square roots and other roots of algebraic expressions. We will also consider their related functions.
Square Roots When solving problems, we must often find what number must be squared to obtain a second number a. If such a number can be found, it is called a square root of a. For example, ⢠0 is a square root of 0, because 02 0. ⢠4 is a square root of 16, because 42 16. ⢠4 is a square root of 16, because 2 16. ⢠7xy is a square root of 49x 2y 2, because 2 49x 2y 2. ⢠7xy is a square root of 49x 2y 2, because 2 49x 2y 2. All positive numbers have two real-number square roots: one that is positive and one that is negative.
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Exam Ple 2 Refer To Figure 2
Analyze the problem In Figure 2-6, we have two angles that are side by side. From the figure, we can see that the sum of their measures is 75°.
Form an equation Since the sum of x and 37° is equal to 75°, we can form the equation as follows. The angle that the angle that the angle that plus equals measures x measures 37° measures 75°. x 37 75
Solve The Equation We Can Solve This Equation As Follows:
State the conclusion The dimensions of the dog run are 4 meters by 10 meters.
Check the result If the dog run has a width of 4 meters and a length of 10 meters, its length is 6 meters longer than its width, and the perimeter is 2 2 28. â
Inequalities Recall that inequalities are statements indicating that quantities are unequal. ⢠a b means âa is less than b.â ⢠a b means âa is greater than b.â 7.1 Review of Equations and Inequalities 433
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To Find The Inverse Of A Function Find The Inverse Of Each Function
56. y 2x 2 1
57. y 0 x 0
Chapter TestSolve each equation by factoring. 13. Graph Æ 12 x 2 4 and y give the coordinates of its 1. x 3x 18 0 2
Determine what number must be added to each binomialto make it a perfect square. 3. x 2 24x 4. x 2 50x 14. Graph: y x 2 3. ySolve each equation by completing the square. 5. x 2 4x 1 0 6. x 2 5x 3 0
The Graphing Method 1 Carefully Graph Each Equation
2x 3y 2 EXAM PLE 1 Use graphing to solve e . 3x 2y 16
Solution Using the intercept method, we graph both equations on one set of coordinate axes, as shown in Figure 3-22. 3.3 Solving Systems of Equations by Graphing 177
2x 3y 2 3x 2y 16 3x = 2y + 16
x y x y 1 0, 2 2 2 0 8 0 1 163, 0 2 0 3 3 16 2x + 3y = 2 1 0 3 2 2 4 2
Figure 3-22
Although there are infinitely many pairs that satisfy 2x 3y 2 and infi- nitely many pairs that satisfy 3x 2y 16, only the coordinates of the point where the graphs intersect satisfy both equations. The solution is x 4 and y 2, or just . To check, we substitute 4 for x and 2 for y in each equation and verify that the pair satisfies each equation.
2x 3y 2 3x 2y 16 2 3 2 3 2 16 8 6 2 12 4 16 2 2 12 12
The equations in this system are independent equations, and the system is a consis- tent system of equations.
Self Check Use graphing to solve e . â x y 1
Sometimes a system of equations will have no solution. These systems are called in- consistent systems.
EXAM PLE 2 Solve the system: e . 4x 2y 8
Solution We graph both equations on one set of coordinate axes, as in Figure 3-23. 178 Chapter 3 Graphing and Solving Systems of Equations and Inequalities
2x + y = â6 2x y 6 4x 2y 8 4x + 2y = 8 x y x y 3 0 2 0 x 0 6 0 4 2 2 1 2
Figure 3-23
2y 3x Self Check Solve the system: e . â 3x 2y 6
y 2x 4 EXAM PLE 3 Solve the system: e . 4x 8 2y
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Vocabulary And Concepts Fill In The Blanks
x17. Carpentry The 12-foot board in the illustration has been cut into two parts, one twice as long as the other. 22. Football In 1967, Green Bay beat Kansas City by How long is each part? 25 points in the first Super Bowl. If a total of 45 points were scored, what was the final score of the game? GEOMETRY In Exercises 23â30, find x. 12 ft18. Plumbing A 20-foot pipe has been cut into two 123° parts, one 3 times as long as the other. How long is x each part? 50° x19. Robotics If the robotic arm shown in the illustra- 40° tion will extend a total distance of 30 feet, how long 25. 26.
114 Chapter 2 Equations and Inequalities
Solution By Factoring Solution By Formula
cm Since the rectangle cannot have a negative width, we discard the solution of 23. Thus, the only solution is w 11. Since the rectangle is 11 centimeters wide and centimeters long, its dimensions are 11 centimeters by 23 centimeters.
Check: 23 is 12 more than 11, and the area of a rectangle with dimensions of Figure 10-2 23 centimeters by 11 centimeters is 253 square centimeters. â 656 Chapter 10 Quadratic Functions, Inequalities, and Algebra of Functions
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The Graph Of 2x K Is The Graph Of 1x Translated K Units To
EXAM PLE 8 Graph Æ 2x 4 2 and find its domain and range.
Solution This graph will be the reflection of Æ 1x about the x-axis, translated 4 units to the left and 2 units down. See Figure 9-2. We can confirm this graph by using a graphing calculator with window settings of for x and for y to get the graph shown in Figure 9-2.
â4
From either graph, we can see that the domain is the interval .
Solve Each System Of Equations Algebraically For Real
x 2 y 2 10 x2 y2 59. e 10. e x2 y2 2 x 2 y 2 36 y 3x 2 x y 3 19. e 20. e y y x y 2 49x 2 36y 2 1,764
x2 y2 5 x2 x y 2 21. e 22. e x y 3 4x 3y 0 x x
x 2 y 2 13 x 2 y 2 25 23. e 24. e y x2 1 2x 2 3y 2 5
x 2 y 2 25 x 2 y 2 1311. e 12. e 12x 2 64y 2 768 y x2 1 x 2 y 2 30 9x 2 7y 2 81 y y 25. e 26. e y x2 x2 y2 9
x 2 y 2 13 2x 2 y 2 6 27. e 28. e x x x2 y2 5 x2 y2 3
9 x 2 y 2 20 xy x 13 y 29. e 2 30. ⢠213. e 14. e x y 2 12 y 2x 4 y x2 3x 2y 6 y y
y 2 40 x 2 x 2 6x y 5 31. e 32. e y x 2 10 x 2 6x y 5 x x
y x2 4 6x 2 8y 2 182 33. e 34. e x 2 y 2 16 8x 2 3y 2 24
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Baseball In Baseball The Pitchers Mound Is 60 Feet
9 ft38. Baseball The shortstop fields a grounder at a point one-third of the way from second base to third base. How far will he have to throw the ball to make an out at first base? 43. Telephone service The telephone cable in the illus- tration currently runs from A to B to C to D. How Use a calculator. much cable is required to run from A to D directly?
Check The Solution In The Original Equations
2a 3b 7 Self Check Solve the system: e . â 5a 2b 1
! Comment Note that a solution of Example 3 by the substitution method would involve fractions. In these cases, the addition method is usually easier.
5 2 7 7 9 21 3.5 Solving Systems of Equations by Addition 195
Solution To clear the equations of fractions, we multiply both sides of the first equation by 6 and both sides of the second equation by 63. This gives the system 5x 4y 7 90x 28y 51 We can solve for x by eliminating the terms involving y. To do so, we multiply Equa- tion 1 by 7 and add the result to Equation 2. 35x 28y 49 x Divide both sides by 125. 125 To solve for y, we substitute 45 for x in Equation 1 and simplify. 5x 4y 7
4y 3 Subtract 4 from both sides. 3 y Divide both sides by 4. 4 Check the solution of 1 45, 34 2 in the original equations.
y 6 2
Solution We can multiply both sides of the first equation by 3 and both sides of the second equation by 2 to clear the equations of fractions.
b 3a b
196 Chapter 3 Graphing and Solving Systems of Equations and Inequalities
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To Perform This Multiplication With A Calculator We Enter These Numbers And
To divide decimals, we move the decimal point in the divisor to the right to make the divisor a whole number. We then move the decimal point in the dividend the same number of places to the right. 1.23 30.258 Move the decimal point in both the divisor and the dividend two places to the right.
We align the decimal point in the quotient with the repositioned decimal point in the dividend and use long division. 24.6 To perform the previous division with a calculator, we enter these numbers and press these keys: 30.258 1.23 On a scientific calculator. 30.258 1.23 ENTER On a graphing calculator.
Rounding Decimals We often round long decimals to a specific number of decimal places. For example, the decimal 25.36124 rounded to one place is 25.4. Rounded to two places , the decimal is 25.36. To round deci- mals, we use the following rules.
Rounding Decimals 1. Determine to how many decimal places you wish to round. 2. Look at the first digit to the right of that decimal place. 3. If that digit is 4 or less, drop it and all digits that follow. If it is 5 or greater, add 1 to the digit in the position to which you wish to round, and drop all digits that follow.
EXAM PLE 10 Auto loans Juan signs a one-year note to borrow $8,500 to buy a car. If the rate of interest is 614 % , how much interest will he pay?
Self Check How much interest will Juan pay if the rate is 9%? â
Exam Ple 1 Solve: 3 2x 9
Solution We use the distributive property to remove parentheses and then isolate x on the left- hand side of the equation. 7.1 Review of Equations and Inequalities 429
3 2x 9 6x 3 2x 9 To remove parentheses, use the distributive property. 6x 3 3 2x 9 3 To undo the subtraction by 3, add 3 to both sides. 6x 2x 12 Combine like terms. 6x 2x 2x 12 2x To eliminate 2x from the right-hand side, subtract 2x from both sides. 4x 12 Combine like terms. x 3 To undo the multiplication by 4, divide both sides by 4.
Check: We substitute 3 for x in the original equation to see whether it satisfies the equation. 3 2 3 9 3 6 9 On the left-hand side, do the work in parentheses first. 15 15 Since 3 satisfies the original equation, it is a solution. The solution set is .
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Getting Ready Find Each Value
The graph in Figure 11-1 shows the balance in a bank account in which $10,000 was invested in 2000 at 9% annual interest, compounded monthly. The graph shows that in the year 2010, the value of the account will be approximately $25,000, and in the year 2030, the value will be approximately $147,000. The curve shown in Figure 11-1 is the graph of a function called an exponential function, the topic of this section.
There Were 6 Million Shares Outstanding
Self Check If 60% of the shares outstanding were voted in favor of the proposal, how many shares were voted in favor? â
EXAM PLE 13 Quality control After examining 240 sweaters, a quality-control inspector found 5 with defective stitching, 8 with mismatched designs, and 2 with incorrect labels. What percent were defective? 2.1 Solving Basic Equations 89
Solution Let r represent the percent that are defective. Then the base b is 240 and the amount a is the number of defective sweaters, which is 5 8 2 15. We can find r by using the percent formula. r 240 15 Substitute 240 for b and 15 for a. 240r 15 To undo the multiplication of 240, divide both sides by 240. 240 240 r 6.25% To change 0.0625 to a percent, multiply by 100 and insert a % sign.
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Orals 1 Write Y Ln X As An Exponential Equation
11.4 EXERCISES
REVIEW Write the equation of the required line. 16. In the expression ln x, the base is understood to be . 1. Parallel to y 5x 8 and passing through the origin 17. If a population grows exponentially at a rate r, the time it will take the population to double is given by 2. Having a slope of 9 and a y-intercept of the formula t . 18. The logarithm of a negative number is . 3. Passing through the point and perpendicular to the line y 23 x 12 PRACTICE Use a calculator to find each value, if possible. Express all answers to four 4. Parallel to the line 3x 2y 9 and passing through decimal places. the point 5. Vertical line through the point 19. ln 25.25 20. ln 0.523 6. Horizontal line through the point 21. ln 9.89 22. ln 0.00725 23. log 24. ln Simplify each expression. 25. ln 26. log 2x 3 7. Use a calculator to find y, if possible. Express all 4x 2 9 answers to four decimal places. x 1 x 1 8. 27. ln y 2.3015 28. ln y 1.548 x x 1 x 3x 2 x 4 2 9. 2 29. ln y 3.17 30. ln y 0.837 3x 12 x 4 y 31. ln y 4.72 32. ln y 0.48 1 1 33. log y ln 6 34. ln y log 5 x