4.3.1: On or Off the Line?

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Illustrative Mathematics
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Lesson

Let's interpret the meaning of points in a coordinate plane.

Exercise $\PageIndex{1}$: Which One Doesn't Belong: Lines in the Plane

Which one doesn't belong? Explain your reasoning.

Figure $\PageIndex{1}$: Four graphs, each in the x y plane. Graph A. 3 lines that do not intersect. One line crosses the x axis to the left of the origin and the y axis below the origin. Another line crosses the y axis above the origin and the x axis to the right of the origin. The third line crosses the y axis above the origin and the x axis to the right of the origin. Graph B. 2 intersecting lines. One line crosses the x axis to the left of the origin and the y axis below the origin. Another line crosses the y axis above the origin. Graph C. Three lines that intersect at a single point. One line crosses the y axis above the origin. Another line crosses the x axis to the right of the origin and the y axis below the origin. The third line crosses the x axis to the right of the origin. Graph D. Three lines. There are 3 points of intersection between two lines each. One line crosses the y axis above the origin. Another line crosses the x axis to the left of the origin and the y axis above the origin. The third line crosses the x axis to the right of the origin.

Exercise $\PageIndex{2}$: Pocket Full of Change

Jada told Noah that she has $2 worth of quarters and dimes in her pocket and 17 coins all together. She asked him to guess how many of each type of coin she has.

Here is a table that shows some combinations of quarters and dimes that are worth $2. Complete the table.

Table $\PageIndex{1}$
number of quarters	number of dimes
$0$	$20$
$4$
	$0$
	$5$

Here is a graph of the relationship between the number of quarters and the number of dimes when there are a total of 17 coins.
1. What does Point $A$ represent?
2. How much money, in dollars, is the combination represented by Point $A$ worth?

Figure $\PageIndex{2}$: Graph of points in the x y plane, origin O, with grid. Horizontal axis, number of quarters, scale 0 to 24 by 1s. Vertical axis, number of dimes, scale 0 to 24 by 1s. Points plotted are 0 comma 17, 1 comma 16, 2 comma 15, 3 comma 14, 4 comma 13, 5 comma 12, 6 comma 11, 7 comma 10, 8 comma 9 labeled A, 9 comma 8, 10 comma 7, 11 comma 6, 12 comma 5, 13 comma 4, 14 comma 3, 15 comma 2, 16 comma 1, and 17 comma 0.

3. Is it possible for Jada to have 4 quarters and 13 dimes in her pocket? Explain how you know.

4. How many quarters and dimes must Jada have? Explain your reasoning.

Exercise $\PageIndex{3}$: Making Signs

Clare and Andre are making signs for all the lockers as part of the decorations for the upcoming spirit week. Yesterday, Andre made 15 signs and Clare made 5 signs. Today, they need to make more signs. Each person's progress today is shown in the coordinate plane.

Figure $\PageIndex{3}$: Graph of two lines in the x y plane, origin 0, with grid. Horizontal axis, time in minutes, scale 0 to 110, by 5s. Vertical axis, number of completed signs, scale 0 to 70 by 5s. A line crosses the y axis at 5 and passes through the points A and B. Another line crosses the y axis at 15 and passes through the points C and A. Starting at the origin, point A is 8 unit to the right and 5 units up. Starting at the origin, point B is 15 units to the right and 8 point 5 units up. Starting at the origin, point C is 0 units to the right, and 3 units up. Starting at the origin, point D is 20 units to the right and 12 units up.

Based on the lines, mark the statements as true or false for each person.

Table $\PageIndex{2}$
point	what it says	Clare	Andre
$A$	At 40 minutes, I have 25 signs completed.
$B$	At 75 minutes, I ahve 42 and a half signs completed.
$C$	At 0 minutes, I have 15 signs completed.
$D$	At 100 minutes, I have 60 signs completed.

Are you ready for more?

4 toothpicks make 1 square
7 toothpicks make 2 squares
10 toothpicks make 3 squares

Do you see a pattern? If so, how many toothpicks would you need to make 10 squares according to your pattern? Can you represent your pattern with an expression?

Summary

We studied linear relationships in an earlier unit. We learned that values of $x$ and $y$ that make an equation true correspond to points $(x,y)$ on the graph. For example, if we have $x$ pounds of flour that costs $0.80 per pound and $y$ pounds of sugar that costs $0.50 per pound, and the total cost is $9.00, then we can write an equation like this to represent the relationship between $x$ and $y:$

$0.8x+0.5y=9$

Since 5 pounds of flour costs $4.00 and 10 pounds of sugar costs $5.00, we know that $x=5$, $y=10$ is a solution to the equation, and the point $(5,10)$ is a point on the graph. The line shown is the graph of the equation:

Figure $\PageIndex{5}$: The graph of a line in the x y plane. The line slants downward and right, crosses the y axis at 18, and passes through the point 5 comma 10. Two additional points, 9 comma 16 and 1 comma 14, are labeled on the graph.

Notice that there are two points shown that are not on the line. What do they mean in the context? The point $(1,14)$ means that there is 1 pound of flour and 14 pounds of sugar. The total cost for this is $(0.8\cdot 1+0.5\cdot 14$ or $7.80. Since the cost is not $9.00, this point is not on the graph. Likewise, 9 pounds of flour and 16 pounds of sugar costs $0.8\cdot 9+0.5\cdot 16$ or $15.20, so the other point is not on the graph either.

Suppose we also know that the flour and sugar together weigh 15 pounds. This means that

$x+y=15$

If we draw the graph of this equation on the same coordinate plane, we see it passes through two of the three labeled points:

Figure $\PageIndex{6}$: The graph of two intersecting lines in the x y plane. The first line slants downward and right, crosses the y axis at 18, and passes through the point 5 comma 10. The second line slants downward and to the right and passes through the points 1 comma 14 and 5 comma 10. An additional point, 9 comma 16, is labeled on the graph.

The point $(1,14)$ is on the graph of $x+y=15$ because $1+14=15$. Similarly, $5+10=15$. But $9+16\neq 15$, so $(9,16)$ is not on the graph of $x+y=15$. In general, if we have two lines in the coordinate plane,

The coordinates of a point that is on both lines makes both equations true.
The coordinates of a point on only one line makes only one equation true.
The coordinates of a point on neither line make both equations false.

Practice

Exercise $\PageIndex{4}$

1. Match the lines $m$ and $n$ to the statements they represent:

Figure $\PageIndex{7}$: Two lines in an x y plane. Line m passes through points E, B, and D. Point E is at 13 comma 3. Point B is at 6 comma negative 4. Point D is at 2 comma negative 8. Line n passes through points A, B, and C. Point A is at negative 2 comma 4. Point B is at 6 comma negative 4. Point C is at 12 comma negative 10.

A set of points where the coordinates of each point have a sum of 2
A set of points where the $y$-coordinate of each point is 10 less than its $x$-coordinate

2. Match the labeled points on the graph to statements about their coordinates:

Two numbers with a sum of 2
Two numbers where the $y$-coordinate is 10 less than the $x$-coordinate
Two numbers with a sum of 2 and where the $y$-coordinate is 10 less than the $x$-coordinate

Exercise $\PageIndex{5}$

Here is an equation: $4x-4=4x+\underline{ }$. What could you write in the blank so the equation would be true for:

No values of $x$
All values of $x$
One value of $x$

(From Unit 4.2.6)

Exercise $\PageIndex{6}$

Mai earns $7 per hour mowing her neighbors' lawns. She also earned $14 for hauling away bags of recyclables for some neighbors.

Priya babysits her neighbor’s children. The table shows the amount of money $m$ she earns in $h$ hours. Priya and Mai have agreed to go to the movies the weekend after they have earned the same amount of money for the same number of work hours.

Table $\PageIndex{3}$
$h$	$m$
$1$	$$8.40$
$2$	$$16.80$
$4$	$$33.60$

How many hours do they each have to work before they go to the movies?
How much will each of them have earned?
Explain where the solution can be seen in tables of values, graphs, and equations that represent Priya's and Mai's hourly earnings.

Exercise $\PageIndex{7}$

For each equation, explain what you could do first to each side of the equation so that there would be no fractions. You do not have to solve the equations (unless you want more practice).

$\frac{3x-4}{8}=\frac{x+2}{3}$
$\frac{3(2-4)}{4}=\frac{3+r}{6}$
$\frac{4p+3}{8}=\frac{p+2}{4}$
$\frac{2(a-7)}{15}=\frac{a+4}{6}$

(From Unit 4.2.5)

Exercise $\PageIndex{8}$

The owner of a new restaurant is ordering tables and chairs. He wants to have only tables for 2 and tables for 4. The total number of people that can be seated in the restaurant is 120.

Describe some possible combinations of 2-seat tables and 4-seat tables that will seat 120 customers. Explain how you found them.
Write an equation to represent the situation. What do the variables represent?
Create a graph to represent the situation.

4. What does the slope tell us about the situation?

5. Interpret the $x$ and $y$ intercepts in the situation.

(From Unit 3.5.1)

number of quarters	number of dimes
\(0\)	\(20\)
\(4\)
	\(0\)
	\(5\)

\(h\)	\(m\)
\(1\)	\($8.40\)
\(2\)	\($16.80\)
\(4\)	\($33.60\)

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point	what it says	Clare	Andre
\(A\)	At 40 minutes, I have 25 signs completed.
\(B\)	At 75 minutes, I ahve 42 and a half signs completed.
\(C\)	At 0 minutes, I have 15 signs completed.
\(D\)	At 100 minutes, I have 60 signs completed.