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Elementary Algebra

Katherine Yoshiwara

Section 3.6 Chapter 3 Summary and Review

Subsection Lesson 3.1 Intercepts

  • The intercepts of a line are the points where the graph crosses the axes.
  • The intercepts of a graph often represent the starting or ending values for a particular variable.

  • To find the intercepts of a graph.

    • To find the \(x\)-intercept of a graph:
      Substitute \(0\) for \(y\) in the equation and solve for \(x\text{.}\)
    • To find the \(y\)-intercept of a graph:
      Substitute \(0\) for \(x\) in the equation and solve for \(y\text{.}\)

  • To Graph a Linear Equation Using the Intercept Method.

    1. Find the \(x\)- and \(y\)-intercepts of the graph.
    2. Draw the line through the two intercepts.
    3. Find a third point on the graph as a check. (Choose any convenient value for \(x\) and solve for \(y\text{.}\))

Subsection Lesson 3.2 Ratio and Proportion

  • A ratio is a type of quotient that allows us to compare two numerical quantities.

  • Ratio.

    The ratio of \(a\) to \(b\) is written \(\dfrac{a}{b}\text{.}\)
  • A rate is a ratio that compares two quantities with different units.
  • A proportion is a statement that two ratios are equal. In other words, a proportion is a type of equation in which both sides are ratios.
  • We use cross-multiplying to solve proportions.

  • Property of Proportions.

    \begin{equation*} \blert{\text{If}~~\dfrac{a}{b}=\dfrac{c}{d},~~~\text{then}~~~ad=bc.} \end{equation*}
  • Two variables are said to be proportional if their ratios are always the same.

Subsection Lesson 3.3 Slope

  • The slope of a line measures the rate of change of \(y\) with respect to \(x\text{.}\)

  • Slope.

    The slope of a line is defined by the ratio
    \begin{equation*} \blert{\dfrac{\text{change in } y\text{-coordinate}}{\text{change in } x\text{-coordinate}}} \end{equation*}
    as we move from one point to another on the line. In symbols,
    \begin{equation*} \blert{m=\dfrac{\Delta y}{\Delta x}} \end{equation*}
  • Lines have constant slope. No matter what two points on a line we use to calculate its slope, we always get the same result.
  • The slope of a horizontal line is zero, and the slope of a vertical line is undefined.

Subsection Lesson 3.4 Slope-Intercept Form

  • Slope-Intercept Form.

    A linear equation written in the form
    \begin{equation*} \blert{y=mx+b} \end{equation*}
    is said to be in slope-intercept form. The coefficient \(m\) is the slope of the graph, and\(b\) is the \(y\)-intercept.
  • The constant term in the slope-intercept form tells us the initial value of \(y\text{,}\) and the coefficient of \(x\) tells us the rate of change of \(y\) with respect to \(x\text{.}\)

  • To Graph a Line Using the Slope-Intercept Method.

    1. Write the equation in the form \(y=mx+b\text{.}\)
    2. Plot the \(y\)-intercept, \((0,b)\text{.}\)
    3. Write the slope as a fraction, \(m = \dfrac{\Delta y}{\Delta x}\text{.}\)
    4. Use the slope to find a second point on the graph: Starting at the \(y\)-intercept, move \({\Delta y}\) units in the \(y\)-direction, then \({\Delta x}\) units in the \(x\)-direction.
    5. Find a third point by moving \({-\Delta y}\) units in the \(y\)-direction, then \({-\Delta x}\) units in the \(x\)-direction, starting from the \(y\)-intercept.
    6. Draw a line through the three plotted points.

Subsection Lesson 3.5 Properties of Lines

  • Parallel and Perpendicular Lines.

    1. Two lines are parallel if their slopes are equal, that is, if
      \begin{equation*} \blert{m_1 = m_2} \end{equation*}
      or if both lines are vertical.
    2. Two lines are perpendicular if the product of their slopes is \(-1\text{,}\) that is, if
      \begin{equation*} \blert{m_1 m_2 = -1} \end{equation*}
      or if one of the lines is horizontal and one is vertical.

  • Horizontal and Vertical Lines.

    1. The equation of the horizontal line passing through \((0,b)\) is
      \begin{equation*} \blert{y=b} \end{equation*}
    2. The equation of the vertical line passing through \((a,0)\) is
      \begin{equation*} \blert{x=a} \end{equation*}

  • Two-Point Formula for Slope.

    The slope of the line joining points \(P_1(x_1,y_1)\) and \(P_2(x_2,y_2)\) is
    \begin{equation*} \blert{m=\dfrac{y_2-y_1}{x_2-x_1}}~~~~~~\text{if}~~~~x_2 \not= x_1 \end{equation*}

Subsection Review Questions

Use complete sentences to answer the questions.
  1. Explain how to find the intercepts of a graph.
  2. What is the difference between a ratio and a proportion?
  3. State the fundamental property of proportions.
  4. How can we tell if two variables are proportional?
  5. Give a formula for slope using the notation.
  6. State the slope-intercept formula, and explain the meaning of the coefficients.
  7. Explain the slope-intercept method for graphing a line.
  8. What is the easiest way to find the slope of a line from its equation?
  9. How can we tell if two lines are perpendicular?
  10. What is the difference between a line whose slope is zero and a line whose slope is undefined?

Subsection Review Problems

Exercises Exercises

Exercise Group.
For Problems 1–2, find the \(x\)-and \(y\)-intercepts of the line, then graph the line.
1.
\(6x-4y=12\)
grid
Answer.
\((2,0) \text{,}\) \((0,-3)\)
triangle
2.
\(y=-2x+8\)
grid
3.
Monica has saved $7800 to live on while she attends college. She spends $600 a month.
  1. Write an equation for the amount, \(S\text{,}\) in Monica’s savings account after \(m\) months.
  2. Write and solve an equation to answer the question: How long will it be before her savings are reduced to $2400?
  3. Explain what the intercepts of your equation mean in terms of the problem.
Answer.
  1. \(\displaystyle S=7800-600m\)
  2. \(2400=7800-600m\text{;}\) \(~m=9\) months
  3. The \(S\)-intercept at \(S=7800\) shows that she started with $7800. The \(m\)-intercept at \(m=13\) shows that she will use all the money in 13 months.
4.
Ronen is on a rock-climbing expedition. He is climbing out of a deep gorge at a rate of 4 feet per minute, and right now his elevation is \(-156\) feet.
  1. Write an equation for Ronen’s elevation, \(h\text{,}\) after \(m\) minutes.
  2. Write and solve an equation to answer the question: When will his elevation be \(-20\) feet?
  3. Explain what the intercepts of your equation mean in terms of the problem.
Exercise Group.
For Problems 5–6, write ratios or rates.
5.
In an anthropology class of 35 students, 14 are men. What is the ratio of men to women in the class?
Answer.
\(\dfrac{2}{3} \)
6.
Zach’s car went 210 miles on 8.4 gallons of gasoline, and Tasha’s car went 204 miles on 8 gallons of gasoline. Which car had the higher rate of fuel consumption?
Exercise Group.
For Problems 7–8, solve the proportion.
7.
\(\dfrac{105}{y}=\dfrac{15}{17}\)
Answer.
\(119\)
8.
\(\dfrac{16}{q}=\dfrac{52}{86.125}\)
Exercise Group.
For Problems 9–14, write and solve a proportion to answer the question.
9.
At Van’s Hardware, 36 metric frimbles cost $4.86. How many metric frimbles can you buy for $6.75?
Answer.
\(50\)
10.
Bob’s weekly diet includes 70 grams of fat, 200 grams of protein, and 1800 grams of carbohydrates. Jenni’s weekly diet includes 112 grams of fat. How many grams of protein and carbohydrates should she consume so that her diet has the same proportions as Bob’s diet?
11.
Simon joined a co-op that provides access to the Internet. His user bill, \(B\text{,}\) is proportional to the number of hours, \(h\text{,}\) that he is logged on to the net. Last month he was logged on for 28 hours, and his bill was $21. This month Simon’s bill was $39. How many hours did he log on the Internet?
Answer.
52 hr
12.
A small orchard that is 150 yards on each side contains 100 apple trees. The number of trees in an orchard is proportional to the area of the orchard. How many trees are in a larger orchard that is 600 yards on each side?
13.
On a map of Arenac County, 3 centimeters represents 5 miles.
  1. What are the true dimensions of a rectangular township whose dimensions on the map are 6 centimeters by 9 centimeters?
  2. What is the perimeter of the township? What is the perimeter of the corresponding region on the map?
  3. What is the ratio of the perimeter of the actual township to the perimeter of the corresponding region on the map?
  4. What is the area of the township? What is the area of the corresponding region on the map?
  5. What is the ratio of the area of the actual township to the area of the corresponding region on the map?
Answer.
  1. 10 mi by 15 mi
  2. 50 mi; 30 cm
  3. \(\displaystyle \dfrac{5}{3} \)
  4. 150 sq mi; 54 sq cm
  5. \(\displaystyle \dfrac{25}{9} \)
14.
On a map of Euclid County, \(\dfrac{1}{3}\) inch represents 2 miles. Lake Pythagoras is represented on the map by a right triangle with sides 1 inch, \(\dfrac{4}{3}\) inches, and \(\dfrac{5}{3}\) inches.
  1. What are the true dimensions of Lake Pythagoras?
  2. What is the perimeter of Lake Pythagoras? What is the perimeter of the corresponding region on the map?
  3. What is the ratio of the perimeter of the actual lake to the perimeter of the corresponding region on the map?
  4. What is the area of Lake Pythagoras? What is the area of the corresponding region on the map?
  5. What is the ratio of the area of the actual lake to the are of the corresponding region on the map?
Exercise Group.
For Problems 15–16, decide whether the two variables are proportional.
15.
  1. \(D\) \(0.5\) \(1\) \(2\) \(4\)
    \(V\) \(0.25\) \(2\) \(16\) \(64\)
  2. \(s\) \(0.2\) \(0.8\) \(1.6\) \(2.5\)
    \(M\) \(0.08\) \(0.32\) \(0.64\) \(1\)
Answer.
  1. No
  2. Yes
16.
  1. Time \(5\) \(10\) \(15\) \(20\)
    Cost \(1.50\) \(2.50\) \(3.50\) \(4.50\)
  2. Speed \(30\) \(40\) \(50\) \(60\)
    Distance \(42\) \(56\) \(70\) \(84\)
Exercise Group.
For Problems 17–18, use the graph to find the slope of each line, and illustrate \(\Delta x\) and \(\Delta y\) on the graph.
17.
\(3x+2y=-7\)
graph of line
Answer.
\(\dfrac{-3}{2} \)
18.
\(3y=5x\)
graph of ine
19.
The Palm Springs aerial tramway ascends Mt. San Jacinto at a slope of approximately 0.516. It traverses a horizontal distance of about 11,380 feet. The elevation at the bottom of the tramway is 2643 feet. What is the elevation at the top?
Answer.
8515 ft
20.
The ruins of the Pyramid of Cholula in Guatemala have a square base 1132 feet on each side. The sides of the pyramid rise at a slope of about 0.32. How tall was the pyramid originally?
Exercise Group.
For Problems 21–24,
  1. Find the intercepts of the line.
  2. Use the intercepts to graph the line.
  3. Use the intercepts to find the slope of the line.
21.
\(6x-3y=18\)
grid
Answer.
  1. \((0,-6) \text{,}\) \(~(3,0) \)
  2. line
  3. \(\displaystyle m=2\)
22.
\(4y+9x=36\)
grid
23.
\(\dfrac{x}{2}-\dfrac{y}{3}=1\)
grid
Answer.
  1. \((2,0) \text{,}\) \(~(0,-3) \)
  2. line
  3. \(\displaystyle m=\dfrac{3}{2} \)
24.
\(y=3x-8\)
grid
Exercise Group.
For Problems 25–26,
  1. Find the slope and \(y\)-intercept of the line.
  2. Write an equation for the line.
25.
graph of line
Answer.
  1. \(m=\dfrac{-3}{5} \text{,}\) \(~b=3\)
  2. \(\displaystyle y=\dfrac{-3}{5}x+3 \)
26.
graph of line
Exercise Group.
For Problems 27–30,
  1. Find the slope and \(y\)-intercept of the line.
  2. Graph the line using the slope-intercept method.
27.
\(2y+5x=-10\)
grid
Answer.
  1. \(\dfrac{-5}{2} \text{,}\) \(~b=-5\)
  2. line
28.
\(y+3x=0\)
grid
29.
\(3x=4y-8\)
grid
Answer.
  1. \(\dfrac{3}{4} \text{,}\) \(~b=2\)
  2. line
30.
\(\dfrac{x}{4}-\dfrac{y}{5}=1\)
grid
31.
This year Francine bought a 3-foot tall blue spruce sapling for her front yard, and it is supposed to grow about 6 inches per year. Write an equation for the height, \(h\text{,}\) of the tree \(t\) years from now.
Answer.
\(h=36+6t\)
32.
The price of a medium bowl of frozen yogurt is given by the equation
\begin{equation*} y=1.35+0.85t \end{equation*}
where \(t\) is the number of toppings you select. Find the slope and the \(y\)-intercept of the graph, and explain what each means in terms of the problem.
33.
Beryl is sailing in a hot air balloon. Her altitude at \(t\) minutes after 2 pm is given in feet by
\begin{equation*} h=500-15t \end{equation*}
Find the slope and the \(h\)-intercept of the graph, and explain what each means in terms of the problem.
Answer.
\(m=-15\text{:}\) Beryl is descending at a rate of 15 ft/sec.
\(b=500\text{:}\) At 2 pm, Beryl’s altitude was 500 ft.
34.
The amount of water in the municipal swimming pool is given in gallons by
\begin{equation*} y=500,000-5000h \end{equation*}
where \(h\) is the number of hours since they started draining the pool for the winter. State the slope and \(y\)-intercept of the equation, and explain their meaning in terms of the problem.
Exercise Group.
For Problems 35–36, graph the line with the given slope and passing through the given point.
35.
\(m=-\dfrac{3}{4},~~(2,-1)\)
grid
Answer.
line
36.
\(m=\dfrac{1}{3},~~(0,-3)\)
grid
Exercise Group.
For Problems 37–40, find the slope of the line segment joining the points.
37.
\((-1,4),~ (3,-2)\)
Answer.
\(\dfrac{-3}{2} \)
38.
\((5,0),~ (2,-6)\)
39.
\((6.2,1.4),~ (-2.1,4.8)\)
Answer.
\(-0.4\)
40.
\((0,-6.4),~ (-5.6,3.2)\)
Exercise Group.
For Problems 41–42, find the slope of the line described.
41.
Vertical with \(x\)-intercept \((-5,0)\)
Answer.
undefined
42.
Perpendicular to the line \(~y=\dfrac{1}{3}x-2\)
43.
It costs Delbert $50.40 to fill up the 12-gallon gas tank in his sports car, and $105 to fill the 25-gallon tank in his recreational vehicle.
  1. If you plot price, \(p\text{,}\) on the vertical axis, and gallons, \(g\text{,}\) on the horizontal axis, compute the slope of the line segment joining the two points.
  2. What does the slope tell you about the problem?
Answer.
  1. 4.2
  2. Gas costs $4.20 per gallon.
44.
Find the slope and -intercept of the line \(~y=4\text{.}\)
Exercise Group.
For Problems 45–48, find an equation for the line described. Write your answers in slope-intercept form if possible.
45.
With \(x\)-intercept \((5,0)\) and \(y\)-intercept \((0,-1)\)
Answer.
\(y=\dfrac{1}{5} x-1\)
46.
Passing through the points \((0,-2)\) and \((3,-5)\)
47.
Parallel to the \(y\)-axis and passing through the point \((-4,2)\)
Answer.
\(x=-4\)
48.
Horizontal and passing through the point \((-3,8)\)
Exercise Group.
For Problems 49–50, decide whether the lines are parallel, perpendicular, or neither.
49.
  1. \(\displaystyle y=\dfrac{1}{2}x+3;~~x-2y=8\)
  2. \(\displaystyle 4x-y=6;~~x+4y=-2\)
Answer.
  1. parallel
  2. perpendicular
50.
  1. \(\displaystyle 5x+3y=1;~~5y-3x=6\)
  2. \(\displaystyle 4y-5x=1;~~2y-\dfrac{5}{2}=2\)
Exercise Group.
For Problems 51–58, use the most convenient method to graph the equation.
51.
\(4x-3y=12\)
Answer.
line
52.
\(\dfrac{x}{6}-\dfrac{y}{12}=1\)
53.
\(50x=40y-20,000\)
Answer.
line
54.
\(1.4x+2.1y=8.4\)
55.
\(3x-4y=0\)
Answer.
line
56.
\(x=-4y\)
57.
\(4x=-12\)
Answer.
line
58.
\(2y-x=0\)
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