Chapter 19 Heart of Algebra

3(1/2-y)=3/5+15y What is the solution to the equation above?

Distributing the left side would give you 3/2-3y=3/5+15y which can be rewritten as 18y=3/2-3/5. Multiply each side by 10, the least common multiple of 2 and 5, clears the denominators: 180y=30/2-30/5=15-6=9. Therefore, y=9/180=1/20

The graph of line k is shown in the xy-plane above. Which of the following is an equation of a line that is perpendicular to line k?

The graph of line k passes through (0,6) and (3,0). Thus, the slope of line k is 0-6/3-0=-2. Perpendicular lines have negative reciprocal slopes, so the only choice that has 1/2 as the slope is y=1/2x+3.

-2(3x-2.4)=-3(3x-2.4)

The structure of the equation reveals that -2 times a quantity, 3x-2.4, is equal to -3 times the same quantity. This is only possible if the quantity 3x-2.4 is equal to zero. Thus, 3x-2.4=0 or 3x=2.4. Therefore, the solution is x=0.8.

In 2014, County X had 783 miles of paved roads. Startimg in 2015, the county has been building 8 miles of new paved roads each year. At this rate, how many miles of paved road will County C have in 2030? (Assume that no paved roads go out of service.)

2030-2014=16 783+8n 783+8(16)=783+128=911 (n=years)

In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. At this rate, in which year will County X first have at least a 1,000 miles of paved roads? (Assume that no paved roads go out of service.)

783+8n≥1,000 -783 -783 8n≥217 8n/8≥217/8 n≥27.125 (note that n is counting calendar years and so it must be an integer) 783+8(27)=999 County X will first have AT LEAST 1,000 miles of paved roads in 28 years after 2014, or 2042.

A voter registration drive was held in Town Y. The number of voters, V, registered T days after the drive began can be estimated by the equation V=3,450+65T. What is the best interpretation of the number 65 in this equation?

For each day that passes, it is the next day of the registration drive, and so T increases by 65. That is, the number of voters registered increased by 65 for each day of the drive. Therefore, 65 is the number of the number of voters registered each day during the drive.

3s-2t=a -15s+bt=-7 In the system of equations above, a and b are constants. If the system has infinitely many solutions, what is the value of a?

If a system of two linear equations in two variables has infinitely many solutions, the two equations in the system must be equivalent. Since the two equations are presented in the same form, the second equation must be equal to the first equation multiplied by a constant. Since the coefficient of s in the second equation is -5 times the coefficient of s in the first equation, multiply each side of the first by -5. This gives -15s+10t=5a, -15s+bt=-7 Since these two equations are equivalent and have the same coefficient of s, the coefficients of t and the constants on the right-hand side must also be the same. Thus, b=10 and -5a=-7. Therefore, the value of a is 7/5.

2y+6x=3 y+3x=2 How many solutions (x,y) are there to the system of equations above?

If you multiply each side of y+3x=2 by 2, you get 2y+6x=4. Then subtracting each side of 2y+6x=3 from the corresponding side of 2y+6x=4 gives 0=1. This is a false statement. Therefore, the system has zero solutions.

Each morning, John jogs at 6 miles per hour and rides a bike at 12 miles per hour. His goal is to jog and ride his bike a total of at least 9 miles in less than 1 hour. If John jogs j miles and rides his bike at b miles, which of the following systems of inequalities represents John's goal?

John jogs j miles and rides his bike b miles; his goal is to jog and ride his bike a total of at least 9 miles is represented by the inequality j+b≥9 Since rate x time=distance, it follows that time is equal to distance divided by rate. So, John's goal can be represented by the inequality j/6+b/12<1. The system j+b≥9 and j/6+b/12<1 would be the correct answer.

To edit a manuscript, Miguel charges $50 for the first 2 hours and $20 per hour after the first 2 hours. Which of the following expresses the amount in dollars, C, Miguel charges of it takes him x hours to edit a manuscript, where x>2?

Note that since the $50 that Miguel charges pays for his first 2 hours of editing, he charges $20 per hour only AFTER the first 2 hours. If it takes x hours for Miguel to edit a manuscript, he charges $50 for the first 2 hours and $20 per hour for the remaining time, which is x-2 hours. his total charge can be written as C=50+20(x-2). Expanded, it is C=50+20x-40, or C=20x+10

Maizah bought a pair of pants and a briefcase at a department store. The sum of the prices before sales tax was $130.00. There was no sales tax on the pants and a 9% sales tax on the briefcase. The total Maizah paid, including the sales tax, was $136.75. What was the price, in dollars, of the pants?

P=price of pants B=price of briefcase P+B=130 B+0.09B=1.09B (price of briefcase with sales tax) P+1.09B=136.75 Now you need to solve the system: (P+1.09B)-(P+B)=136.75-130 simplifies to 0.09B=6.75 Now divide: B=6.75/0.09=75 This is the value of the price of the briefcase Substitute 75 for B in the equation P+B=130 which gives you P+75=130 which equals 55 The pants equal $55.

-2x=4y+6 2(2y+3)=3x-5 What is the solution (x,y) to the system of equations above?

Since -2x=4y+6, it follows that -x=2y+3. Now substitute -x for 2y+3 in the second equation. That gives you 2(-x)=3x-5, which simplifies to 5x=5, or x=1. Substituting 1 for x in the first equation gives you -2=4y+6, which simplifies to 4y=-8, or y=-2. Therefore, the solution to the system is (1,-22).

In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. At this rate, if n is the number of years after 2014, which of the following functions f gives the number of miles of paved road there will be in County X? (Assume that no paved roads go out of service.)

f(n)=783+8n

The stratosphere is the layer of the Earth's atmosphere that is more than 10 kilometers (km) and less than 50km above the Earth's surface. Which of the following inequalities describes all possible heights x, in km, above the Earth's surface that are in the stratosphere?

the possible heights x, in km, above the Earth's surface that are in the stratosphere are given by the inequality 10<x<50. To answer the question, you need to find an absolute value inequality that is equivalent to 10<x<50. The inequality 10<x<50 describes the open interval (10,50). To describe an interval with an absolute value inequality, use the midpoint and size of the interval. The midpoint of (10,50) is 10+50/2=30. Then observe that the interval (10,50) consists of all points that are within 20 of the midpoint. That is, (10,50) consists of x, whose distance from 30 on the number line is less than 20. The distance between x and 30 on the number line is |x-30|. Therefore, the possible values of x are described by |x-30|<20.