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Elementary Algebra

Katherine Yoshiwara

Section 4.6 Chapter 4 Summary and Review

Subsection Lesson 4.1 The Distributive Law

  • Distributive Law.

    If \(a,~ b,\) and \(c\) are any numbers, then
    \begin{equation*} \blert{a(b+c) = ab + ac} \end{equation*}

  • Steps for Solving Linear Equations.

    1. Use the distributive law to remove any parentheses.
    2. Combine like terms on each side of the equation.
    3. By adding or subtracting the same quantity on both sides of the equation, get all the variable terms on one side and all the constant terms on the other.
    4. Divide both sides by the coefficient of the variable to obtain an equation of the form \(x=a\text{.}\)

Subsection Lesson 4.2 Systems of Linear Equations

  • A pair of linear equations in two variables
    \begin{gather*} a_1x+b_1y=c_1\\ a_2x+b_2y=c_2 \end{gather*}
    considered together is called a system of linear equations, or a linear system.
  • A solution to a system of linear equations is an ordered pair \((x,y)\) that satisfies each equation in the system. It is the intersection point of the two lines described by the system.
  • There are three types of linear systems:
    1. Consistent and independent system. The graphs of the two lines intersect in exactly one point. The system has exactly one solution.
    2. Inconsistent system. The graphs of the equations are parallel lines and hence do not intersect. An inconsistent system has no solutions.
    3. Dependent system. All the solutions of one equation are also solutions to the second equation, and hence are solutions of the system. The graphs of the two equations are the same line. A dependent system has infinitely many solutions.

Subsection Lesson 4.3 Algebraic Solution of Systems

  • To Solve a System by Substitution.

    1. Choose one of the variables in one of the equations. (It is best to choose a variable whose coefficient is 1 or \(-1\text{.}\)) Solve the equation for that variable.
    2. Substitute the result of Step 1 into the other equation. This gives an equation in one variable.
    3. Solve the equation obtained in Step 2. This gives the solution value for one of the variables. Substitute this value into the result of Step 1 to find the solution value of the other variable.

  • To Solve a System by Elimination.

    1. Write each equation in the form \(Ax+By = C\text{.}\)
    2. Decide which variable to eliminate. Multiply each equation by an appropriate constant so that the coefficients of that variable are opposites.
    3. Add the equations from Step 2 and solve for the remaining variable.
    4. Substitute the value found in Step 3 into one of the original equations and solve for the other variable.

  • When Using Elimination to Solve a System.

    1. If combining the two equations results in an equation of the form
      \begin{equation*} 0x+0y=k~~~~(k \not= 0) \end{equation*}
      then the system is inconsistent.
    2. If combining the two equations results in an equation of the form
      \begin{equation*} 0x+0y=0 \end{equation*}
      then the system is dependent.

Subsection Lesson 4.4 Applications of Systems

  • We can use tables to organize the information in applications involving interest, mixtures, or motion.
  • The formula for calculating interest is
    \begin{equation*} \blert{I=Prt} \end{equation*}
    \(I\) is the interest you will earn after \(t\) years if you invest a principal of \(P\) dollars in an account that earns simple interest at an annual rate \(r\text{.}\)
  • The formula for calculating percents is
    \begin{equation*} \blert{P-rW} \end{equation*}
    \(P\) stands for the part obtained when we take \(r\) percent of a whole amount, \(W\text{.}\)
  • In general, we cannot add percents unless they are percents of the same whole amount.
  • To solve problems involving motion at a constant speed we use the formula \blert{D=RT} \(D\) stands for the distance you travel at speed \(R\) in time \(T\text{.}\)

Subsection Lesson 4.5 Point-Slope Form

  • Point-Slope Formula.

    To find an equation for the line of slope \(m\) passing through the point \((x_1,y_1)\text{,}\) we use the point-slope formula
    \begin{equation*} \blert{\dfrac{y-y_1}{x-x_1} = m} \end{equation*}
    or
    \begin{equation*} \blert{y-y_1 = m(x-x_1)} \end{equation*}

  • To Fit a Line through Two Points.

    1. Compute the slope between the two points.
    2. Substitute the slope and either point into the point-slope formula.

Subsection Review Questions

Use complete sentences to answer the questions.
  1. State the two-point formula for slope.
  2. How can you find the equation of a line when you know the slope of the line and one point on the line?
  3. How can you find the solution to a linear system by graphing?
  4. Name two algebraic methods for solving linear systems.
  5. Suppose you are using the elimination method to solve a system. How can you tell if the system is dependent or inconsistent?
  6. How would you label the columns when making a table for a problem about interest?
  7. How would you label the columns when making a table for a problem about a mixture?
  8. How would you label the columns when making a table for a problem about motion?
  9. Under what conditions can we add percents?
  10. Explain why the distributive law does not apply to the expression \(-3(2ab)\text{.}\)

Subsection Review Problems

Exercises Exercises

Exercise Group.
For Problems 1–2, simplify. Which product requires the distributive law?
1.
  1. \(\displaystyle -5(-6m)\)
  2. \(\displaystyle -5(-6-m)\)
Answer.
  1. \(\displaystyle 30m\)
  2. \(-30+5m~\) requires the distributive law.
2.
  1. \(\displaystyle 9(-3-w)\)
  2. \(\displaystyle 9(-3w)\)
Exercise Group.
For Problems 3–6, simplify.
3.
\((4m+2n)-(2m-5n)\)
Answer.
\(2m+7n\)
4.
\((-5c-6)+(-11c+15)\)
5.
\(-7w-2(4w-13)\)
Answer.
\(-15w+26\)
6.
\(4(3z-10)+5(-z-6)\)
Exercise Group.
For Problems 7–10, solve.
7.
\(5p+10(17-p)=2p-5\)
Answer.
\(p=25\)
8.
\(-3(k-2)-4(2k+5)=10+3k\)
9.
\(4(3a-7) \lt -18+2a\)
Answer.
\(a \lt 1\)
10.
\(\dfrac{2x-1}{x+3}=\dfrac{3}{2}\)
11.
The length of a rectangle is 3 times its width.
  1. If the width of a rectangle is \(x\text{,}\) what is its length?
  2. Express the perimeter of the rectangle in terms of \(x\text{.}\)
  3. Suppose the perimeter of the rectangle is 48 centimeters. Find the dimensions of the rectangle.
Answer.
  1. \(\displaystyle 3x\)
  2. \(\displaystyle 8x\)
  3. 6 cm by 18 cm
12.
Last Saturday, a total of 620 people attended the Gaslamp Theater at its two performances.
  1. If \(p\) people attended the matinee last Saturday, how many attended the evening performance?
  2. This Saturday, attendance at the matinee increased by 5%, and attendance at the evening performance increased by 8%. Write expressions in terms of \(p\) for the attendance at each performance.
  3. This Saturday, the total attendance was 663 people. How many people attended the matinee last Saturday?
13.
Wheaton Elementary school plans to buy 30 computers. The computers with speakers cost $1200 each, and those without speakers cost $800 each. Let \(x\) represent the number of computers with speakers. Write expressions in terms of for:
  1. The number of computers without speakers.
  2. The total cost of the computers with speakers.
  3. The total cost of the computers without speakers.
  4. The total cost of all 30 computers.
  5. If Wheaton Elementary has $28,800 to spend on the computershow many of each kind can they buy?
Answer.
  1. \(\displaystyle 30-x\)
  2. \(\displaystyle 1200x\)
  3. \(\displaystyle 800(30-x)\)
  4. \(\displaystyle 400x+24,000\)
  5. 12 with speakers and 18 without speakers
14.
The current in the Lazy River flows at 4 miles per hour.
  1. If your motorboat is traveling at speed \(v\) miles per hour, write expressions for your speed traveling upstream and your speed traveling downstream.
  2. You travel upstream for \(\dfrac{3}{2}\) hours and stop for lunch at an island. Write an expression in terms of \(v\) for the distance you traveled upstream.
  3. You return to your starting point downstream in \(\dfrac{1}{2}\) hour. Write an expression in terms of \(v\) for the distance you traveled downstream.
  4. Write an equation and solve it to find the speed, \(v\text{,}\) of your motorboat.
Exercise Group.
For Problems 15–16, decide whether the given point is a solution of the system.
15.
\(\begin{aligned}[t] x+y\amp=8\amp\amp (-2,10)\\ x-y\amp=2 \end{aligned}\)
Answer.
No
16.
\(\begin{aligned}[t] \amp 8x+3y=21\amp\amp (-3,1)\\ \amp 5x=y-16 \end{aligned}\)
Exercise Group.
For Problems 17–18, solve the system by graphing.
17.
\(\begin{aligned}[t] x+y\amp =5\\ 2x-y\amp =4 \end{aligned}\)
grid
Answer.
\((3,2)\)
18.
\(\begin{aligned}[t] \amp x-y=7\\ \amp y=\dfrac{-2}{3}x-2 \end{aligned}\)
grid
Exercise Group.
For Problems 19–20, solve by substitution.
19.
\(\begin{aligned}[t] \amp y=2x+1\\ \amp 2x+3y=-21\end{aligned}\)
Answer.
\((-3,-5)\)
20.
\(\begin{aligned}[t] x+4y\amp =1\\ 2x+3y\amp =-3 \end{aligned}\)
Exercise Group.
For Problems 21–22, solve by elimination.
21.
\(\begin{aligned}[t] 2x+7y\amp =-19\\ 5x-3y\amp =14 \end{aligned}\)
Answer.
\((1,-3)\)
22.
\(\begin{aligned}[t] 4x+3y\amp =-19\\ 5x+15\amp =-2y \end{aligned}\)
Exercise Group.
For Problems 23–26, solve using substitution or elimination.
23.
\(\begin{aligned}[t] x+5y\amp =18\\ x-y\amp =-3 \end{aligned}\)
Answer.
\(\left(\dfrac{1}{2}, \dfrac{7}{2}\right)\)
24.
\(\begin{aligned}[t] x+5y\amp =11\\ 2x+3y\amp =8 \end{aligned}\)
25.
\(\begin{aligned}[t] \dfrac{2}{3}x-3y\amp =8\\ x+\dfrac{3}{4}y\amp =12 \end{aligned}\)
Answer.
\((12,0)\)
26.
\(\begin{aligned}[t] 3x\amp =5y-6\\ 3y\amp =10-11x \end{aligned}\)
Exercise Group.
For Problems 27–30, decide whether the system is inconsistent, dependent, or consistent and independent.
27.
\(\begin{aligned}[t] 2x-3y\amp =4\\ x+2y\amp =7 \end{aligned}\)
Answer.
Consistent
28.
\(\begin{aligned}[t] 2x-3y\amp =4\\ 6x-9y\amp =4 \end{aligned}\)
29.
\(\begin{aligned}[t] 2x-3y\amp =4\\ 6x-9y\amp =12 \end{aligned}\)
Answer.
Dependent
30.
\(\begin{aligned}[t] x-y\amp =6\\ x+y\amp =6 \end{aligned}\)
Exercise Group.
For Problems 31–42, solve by writing and solving a system of equations.
31.
A health food store wants to produce 30 pounds of granola worth 80 cents per pound. They plan to mix cereal worth 65 cents per pound with dried fruit worth 90 cents per pound. How much of each should they use?
Answer.
12 lb of cereal and 18 lb of dried fruit
32.
The perimeter of a rectangle is 50 yards and its length is 9 yards greater than its width. Find the dimensions of the rectangle.
33.
Last year Veronica made $93 in interest from her savings accounts, one of which paid 6% interest, and the other paid 9%. This year her interest rates dropped to 4% and 8%, respectively, and she made $76 interest. How much does Veronica have invested in each account?
Answer.
$500 in the first account and $700 in the second
34.
Marvin invested $300 more at 6% than he invested at 8%. His total annual income from his two investments is $242. How much did he invest at each rate?
35.
How many pounds of an alloy containing 60% copper must be melted with an alloy containing 20% copper to obtain 8 pounds of an alloy containing 30% copper?
Answer.
2 lb of the 60% alloy and 6 lb of the 20% alloy
36.
Jerry Glove came to bat only 20 times in the first half of the season and got hits 15% of the time. During the second half of the season, Jerry came to bat 140 times and got hits 35% of the time.
  1. How many hits did Jerry get in the first half of the season? How many hits did he get in the second half?
  2. How many times did Jerry come to bat all season? How many hits did he get?
  3. What percent of Jerry’s at-bats resulted in hits?
37.
Alida and Steve are moving to San Diego. Alida is driving their car, and Steve is driving a rental truck. They start together, but Alida drives twice as fast as Steve. After 3 hours they are 93 miles apart. How fast is each traveling?
Answer.
Alida: 62 mph, Steve: 31 mph
38.
Jake rides for the Pony Express, covering his route in 6 hours and returning home, 8 miles per hour slower, in 9 hours. How far does Jake ride?
39.
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?
Answer.
26
40.
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?
41.
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%
42.
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 paying 15% interest to keep his promise?
43.
  1. Graph the line of slope \(\dfrac{-5}{3}\) that passes through the point \((-2,1)\text{.}\)
  2. Find an equation in point-slope form for the line in part (a).
grid
Answer.
  1. line
  2. \(\displaystyle y=\dfrac{-5}{3}x-\dfrac{7}{3}\)
44.
  1. Graph the line of slope \(\dfrac{6}{5}\) that passes through the point \((-3,-4)\text{.}\)
  2. Find an equation in point-slope form for the line in part (a).
grid
Answer.
Exercise Group.
For Problems 45–46, find an equation for the line passing through the two points.
45.
\((3,-5),~(-2,4)\)
Answer.
\(y=\dfrac{-9}{5}x+\dfrac{2}{5}\)
46.
\((0,8),~(4,-2)\)
47.
An interior decorator bases her fee on the cost of a remodeling job. The table below shows her fee, \(F\text{,}\) for jobs of various costs, \(C\text{,}\) both given in dollars.
\(C\) 5000 10,000 20,000 50,000
\(F\) 1000 1500 2500 5500
  1. Write a linear equation for \(F\) in terms of \(C\text{.}\)
  2. Give the slope of the graph, and explain the meaning of the slope in terms of the decorator’s fee.
Answer.
  1. \(\displaystyle F=500+0.10C\)
  2. \(m=0.10~\) The decorator charges 10% of the cost of the job (plus a $500 flat fee)>
48.
Auto registration fees in Connie’s home state depend on the value of the automobile. The table below shows the registration fee \(R\) for a car whose value is \(V\text{,}\) both given in dollars.
\(V\) 5000 10,000 15,000 20,000
\(R\) 135 235 335 435
  1. Write a linear equation for \(R\) in terms of \(V\text{.}\)
  2. Give the slope of the graph, and explain the meaning of the slope in terms of the registration fee.
Exercise Group.
For Problems 49–50,
  1. Make a table of values showing two data points.
  2. Find a linear equation relating the variables.
  3. State the slope of the line, including units, and explain its meaning in the context of the problem.
49.
The population of Maple Rapids was 4800 in 1982 and had grown to 6780 by 1997. Assume that the population increases at a constant rate. Express the population \(P\) of Maple Rapids in terms of the number of years \(t\) since 1982.
Answer.
  1. \(t\) \(0\) \(15\)
    \(P\) \(4800\) \(6780\)
  2. \(\displaystyle P=4800+132t\)
  3. \(m=132~\) people per year gives the rate of population growth
50.
Cicely’s odometer read 112 miles when she filled up her 14-gallon gas tank and 308 when the gas gauge read half full. Express her odometer reading \(m\) in terms of the amount of gas \(g\) she used.
Answer.
51.
  1. Write an equation for any line that is parallel to \(2y=5x-3\text{.}\)
  2. Write an equation for any line that is perpendicular to \(2y=5x-3\text{.}\)
Answer.
  1. \(\displaystyle y=\dfrac{5}{2}x+b\)
  2. \(\displaystyle y=\dfrac{-2}{5}x+b\)
52.
  1. Write an equation for the vertical line that passes through \((3,6)\text{.}\)
  2. Write an equation for the horizontal line that passes through \((3,6)\text{.}\)
53.
Write an equation for the line that is parallel to \(2x+3y=6\) and passes through the point \((1,4)\text{.}\)
Answer.
\(y=\dfrac{-2}{3}x+\dfrac{14}{3}\)
54.
Write an equation for the line that is perpendicular to \(2x+3y=6\) and passes through the point \((1,4)\text{.}\)
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