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Section 2.6 Chapter Summary and Review

Subsection 2.6.1 Glossary

  • scatterplot

  • regression line

  • interpolation

  • extrapolation

  • linear system

  • solution of a system

  • ordered pair

  • inconsistent system

  • dependent system

  • equilibrium price

  • substitution method

  • elimination method

  • linear combination

  • ordered triple

  • back-substitution

  • triangular form

  • Gaussian reduction

Subsection 2.6.2 Key Concepts

  1. We can approximate a linear pattern in a scatterplot using a regression line.

  2. We can use interpolation or extrapolation to make estimates and predictions.

  3. If we extrapolate too far beyond the known data, we may get unreasonable results.

  4. A solution to a \(2\times 2\) linear system is an ordered pair that satisfies both equations.

  5. A solution to a \(2\times 2\) linear system is a point where the two graphs intersect.

  6. The graphs of the equations in an inconsistent system are parallel lines and hence to do not intersect.

  7. The graphs of the two equations in a dependent system are the same line.

  8. If a company's revenue exactly equals its costs (so that their profit is zero), we say that the business venture will break even.

  9. For solving a \(2\times 2\) linear system, the substitution method is easier if one of the variables in one of the equations has a coefficient of \(1\) or \(-1\text{.}\)

  10. If a linear combination of the equations in a system results in an equation of the form

    \begin{align*} 0x+0y=k k \amp\amp(k\ne 0) \end{align*}

    then the system is inconsistent. If an equation of the form

    \begin{gather*} 0x+0y=0 \end{gather*}

    results, then the system is dependent

  11. The point-slope form is useful when we know the rate of change and one point on the line.

  12. The solutions of the linear inequality

    \begin{gather*} ax+by+c\le 0\qquad\text{ or }\qquad ax+by+c\ge 0 \end{gather*}

    consists of the line \(ax+by+c=0\) and a half-plane on one side of that line.

  13. To Graph an Inequality Using a Test Point.

    1. Graph the corresponding equation to obtain the boundary line.

    2. Choose a test point that does not lie on the boundary line.

    3. Substitute the coordinates of the test point into the inequality.

      1. If the resulting statement is true, shade the half-plane that includes the test point.

      2. If the resulting statement is false, shade the half-plane that does not include the test point.

    4. If the inequality is strict, make the boundary line a dashed line.

  14. The solutions to a system of inequalities includes all points that are solutions to all the inequalities in the system. The graph of the system is the intersectionof the shaded regions for each inequality in the system.

  15. A solution to an equation in three variables is an ordered triple of numbers that satisfies the equation.

  16. To solve a \(3\times 3\) linear system, we use linear combinations to reduce the system to triangular form, and then use back-substitution to find the solutions.

Exercises 2.6.3 Chapter 2 Review Problems

1.

The scatterplot shows the boiling temperature of various substances on the horizontal axis and their heats of vaporization on the vertical axis. (The heat of vaporization is the energy needed to change the substance from liquid to gas at its boiling point.)

scatterplot of boiling point vs heat of vaporization
  1. Use a straightedge to estimate a line of best fit for the scatterplot.

  2. Use your line to predict the heat of vaporization of silver, whose boiling temperature is \(2160\degree\)C.

  3. Find the equation of the regression line.

  4. Use the regression line to predict the heat of vaporization of potassium bromide, whose boiling temperature is \(1435\degree\)C.

Answer.
  1. regression line for boiling point vs heat of vaporization
  2. 235 kilojoules

  3. \(\displaystyle y=0.106x + 4.6\)

  4. \(\displaystyle 156.7\degree\text{C}\)

2.

Archaeopteryx is an extinct creature with characteristics of both birds and reptiles. Only six fossil specimens are known, and only five of those include both a femur (leg bone) and a humerus (forearm bone) The scatterplot shows the lengths of femur and humerus for the five Archaeopteryx specimens. Drawa line of best fit for the data points.

scatterplot of femur vs humerus
  1. Predict the humerus length of an Archaeopteryx whose femur is 40 centimeters

  2. Predict the humerus length of an Archaeopteryx whose femur is 75 centimeters

  3. Use your answers from parts (b) and (c) to approximate the equation of a regression line.

  4. Use your answer to part (d) to predict the humerus length of an Archaeopteryx whose femur is 60 centimeters.

  5. Use your calculator and the given points on the scatterplot to find the least squares regression line. Compare the score this equation gives for part (d) with what you predicted earlier. The ordered pairs defining the data are \((38, 41)\text{,}\) \((56, 63)\text{,}\) \((59, 70)\text{,}\) \((64, 72)\text{,}\) \((74, 84)\text{.}\)

Answer.
scatterplot with regression line
  1. \(45\) cm

  2. \(87\) cm

  3. \(\displaystyle y = 1.2x - 3\)

  4. \(69\) cm

  5. \(y = 1.197x - 3.660\text{;}\) \(68.16\) cm

3.

In 1986 the space shuttle Challenger exploded because of O-ring failure on a morning when the temperature was about \(30\degree\text{F}\text{.}\) Previously there had been 1 incident of O-ring failure when temperature was \(70\degree\text{F}\) and 3 incidents whenthe temperature was \(54\degree\text{F}\text{.}\) Use linear extrapolation to estimate the number of incidents of O-ring failure you would expect when the temperature is \(30\degree\text{F}\text{.}\)

4.

Thelma typed a 19-page technical report in 40 minutes. She required only 18 minutes for an 8-page technical report. Use linear interpolation to estimate how long Thelma would require to type a 12-page technical report.

Answer.

26 min

Exercise Group.

For problems 5–6, solve the system by graphing. Use the ZDecimal window.

5.

\(\begin{aligned}[t] y\amp = -2.9x-0.9\\ y\amp = 1.4-0.6x \end{aligned} \)

Answer.

\((-1,2) \)

6.

\(\begin{aligned}[t] y\amp = 0.6x - 1.94 \\ y\amp = -1.1x + 1.29 \end{aligned} \)

Answer.

\((1.9, -0.8) \)

Exercise Group.

For problems 7–10, solve the system by using substitution or elimination.

7.

\(\begin{aligned}[t] x+5y\amp = 18 \\ x-y\amp = -3 \end{aligned} \)

Answer.

\(\left(\dfrac{1}{2}, \dfrac{7}{2} \right) \)

8.

\(\begin{aligned}[t] x+5y\amp = 11\\ 2x+3y\amp = 8 \end{aligned} \)

Answer.

\((1,2) \)

9.

\(\begin{aligned}[t] \dfrac{2}{3}x-3y\amp = 8 \\ x-\dfrac{3}{4}y\amp = 12 \end{aligned} \)

Answer.

\((1,2) \)

10.

\(\begin{aligned}[t] 3x\amp = 5y-6 \\ 3y\amp = 10-11x \end{aligned} \)

Answer.

\(\left(\dfrac{1}{2}, \dfrac{3}{2} \right)\)

Exercise Group.

For problems 11–14, decide whether the system is inconsistent, dependent, or consistent and independent.

11.

\(\begin{aligned}[t] 2x-3y\amp = 4 \\ x+2y\amp = 7 \end{aligned} \)

Answer.

Consistent and independent

12.

\(\begin{aligned}[t] 2x-3y\amp = 4\\ 6x-9y\amp = 4 \end{aligned} \)

Answer.

Inconsistent

13.

\(\begin{aligned}[t] 2x-3y\amp = 4 \\ 6x-9y\amp = 12 \end{aligned} \)

Answer.

Dependent

14.

\(\begin{aligned}[t] x-y\amp = 6 \\ x+y\amp = 6 \end{aligned} \)

Answer.

Consistent and independent

Exercise Group.

For problems 15–20, solve the system using Gaussian reduction.

15.

\(\begin{alignedat}[t]{5} x\amp{}+{}\amp 3y\amp{}-{}\amp z\amp{}=3 \\ 2x \amp{}-{} \amp y\amp{}+{}\amp 3z\amp{}=1 \\ 3x \amp{}+{} \amp 2y\amp {}+{}\amp z\amp{}=5 \end{alignedat} \)

Answer.

\((2,0,-1) \)

16.

\(\begin{alignedat}[t]{5} x\amp{}+{}\amp y\amp{}+{}\amp z\amp{}=2 \\ 3x \amp{}-{} \amp y\amp{}+{}\amp z\amp{}=4 \\ 2x \amp{}+{} \amp y\amp {}+{}\amp 2z\amp{}=3 \end{alignedat} \)

Answer.

\((2,1,-1) \)

17.

\(\begin{aligned}[t] x+z=5 \\ y-z=-8 \\ 2x+z=7 \end{aligned} \)

Answer.

\((2,-5,3) \)

18.

\(\begin{alignedat}[t]{5} x\amp{}+{}\amp 4y\amp{}+{}\amp 4z\amp{}=0 \\ 3x \amp{}+{} \amp 2y\amp{}+{}\amp z\amp{}= -4 \\ 2x \amp{}-{} \amp 4y\amp {}+{}\amp z\amp{}= -11 \end{alignedat} \)

Answer.

\(\left(2,\dfrac{3}{2},-1\right) \)

19.

\(\begin{alignedat}[t]{5} \dfrac{1}{2}x\amp{}+{}\amp y\amp{}+{}\amp z \amp{}={}\amp 3\\ x \amp{}-{} \amp 2y\amp{}-{}\amp \dfrac{1}{3}z \amp {}={}\amp-5 \\ \dfrac{1}{2}x \amp{}-{} \amp 3y\amp {}-{}\amp \dfrac{2}{3}z\amp{}={}\amp -6 \end{alignedat} \)

Answer.

\((-2,1,3) \)

20.

\(\begin{alignedat}[t]{5} \dfrac{3}{4}x\amp{}-{}\amp \dfrac{1}{2}y \amp{}+{}\amp 6z \amp{}={}\amp 2 \\ \dfrac{1}{2}x \amp{}+{} \amp y\amp{}-{}\amp \dfrac{3}{4}z \amp {}={}\amp0 \\ \dfrac{1}{4}x \amp{}+{} \amp \dfrac{1}{2}y\amp {}-{}\amp \dfrac{1}{2}z\amp{}={}\amp 0 \end{alignedat} \)

Answer.

\((2,-1,0) \)

Exercise Group.

Solve Problems 21–26 by writing and solving a system of linear equations in twoor three variables.

21.

A math contest exam has 40 questions. A contestant scores 5 points for each correct answer, but loses 2 points for each wrong answer. Lupe answered all the questions and her score was 102. How many questions did she answer correctly?

22.

A game show contestant wins $25 for each correct answer he gives but loses $10 for each incorrect response. Roger answered 24 questions and won $355. How many answers did he get right?

23.

Barbara wants to earn $500 a year by investing $5000 in two accounts, a savings plan that pays 8% annual interest and a high-risk option that pays 13.5% interest. How much should she invest in each account?

Answer.

$3181.82 at 8%, $1818.18 at 13.5%

24.

An investment broker promises his client a 12% return on her funds. If the broker invests $3000 in bonds paying 8% interest, how much must he invest in stocks, with an estimated rate of return equal to 15% interest, to keep his promise?

Answer.

$4000

25.

The perimeter of a triangle is 30 centimeters. The length of one side is 7 centimeters shorter than the second side, and the third side is 1 centimeter longer than the second side. Find the length of each side.

Answer.

5 cm, 12 cm, 13 cm

26.

A company ships its product to three cities: Boston, Chicago, and Los Angeles. The cost of shipping is $10 per crate to Boston, $5 per crate to Chicago, and $12 per crate to Los Angeles. The company’s shipping budget for April is $445. It has 55 crates to ship, and demand for their product is twice as high in Boston as in Los Angeles. How many crates should the company ship to each destination?

Answer.

20 to Boston, 25 to Chicago, 10 to Los Angeles

Exercise Group.

For Problems 27–30, graph the inequality.

29.

\(y \lt \dfrac{-1}{2}\)

Exercise Group.

For Problems 31–34, graph the solutions to the system of inequalities.

31.

\(y \gt 3, \quad x\le2\)

32.

\(y \ge x, \quad x\gt 2\)

33.

\(3x-6 \lt 6, \quad x+2y\gt 6\)

34.

\(x-3y \gt 3, \quad y\lt x+2\)

Exercise Group.

For Problems 35–38

  1. Graph the solutions to the system of inequalities.

  2. Find the coordinates of the vertices.

35.

\(\begin{aligned}[t] \amp 3x-4y \le 12\\ \amp x\ge0, \quad y\le 0 \end{aligned}\)

36.

\(\begin{aligned}[t] \amp x-2y\le 6\\ \amp y\le x\\ \amp x\ge 0, \quad y\ge 0 \end{aligned}\)

37.

\(\begin{aligned}[t] \amp x+y \le5\\ \amp y\ge x\\ \amp y\ge 2, \quad x\ge 0 \end{aligned}\)

38.

\(\begin{aligned}[t] \amp x-y \le -3\\ \amp x+y \le 6\\ \amp x \le4 \\ \amp x\ge0, \quad y\ge 0 \end{aligned}\)

39.

Ruth wants to provide cookies for the customers at her video rental store. It takes 20 minutes to mix the ingredients for each batch of peanut butter cookies and 10 minutes to bake them. Each batch of granola cookies takes 8 minutes to mix and 10 minutes to bake. Ruth does not want to use the oven more than 2 hours a day, or to spend more than 2 hours a day mixing ingredients. Write a system of inequalities for the number of batches of peanut butter cookies and of granola cookies that Ruth can make in one day, and graph the solutions.

Answer.

\(20p+9g \le 120\text{,}\) \(10p+10g \le 120\text{,}\) \(p\ge 0\text{,}\) \(g\ge 0\)

system of inequalities

40.

A vegetarian recipe calls for 32 ounces of a combination of tofu and tempeh. Tofu provides 2 grams of protein per ounce and tempeh provides 1.6 grams of protein per ounce. Graham would like the dish to provide at least 56 grams of protein. Write a system of inequalities for the amount of tofu and the amountof tempeh for the recipe, and graph the solutions.

Answer.

\(x+y \le 32\text{,}\) \(2x+1.6y \ge 56\text{,}\) \(x\ge 0\text{,}\) \(y\ge 0\text{,}\) where \(x\) represents ounces of tofu, \(y\) the ounces of tempeh

system of inequalities
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