Room for Both

Before you begin

This quiz is about straight lines on a grid, and what they say. A straight line on a graph describes something that changes by the same amount at every step: money paid out grant by grant, a tissue growing back millimeter by millimeter. The quiz covers slope, which is how steep a line is and which way it goes; the equation of a line; the two places where a line crosses the axes; lines that are parallel or perpendicular; and systems, which means two lines at once and the point where they meet. That is family 13 of the test.

The Coordinate Plane starts from the very beginning: the number line, the two axes, and how to find a point on the grid. If the grid itself is new to you, start there. Lines on the Grid then reviews this whole family, and this quiz takes it from there.

The examples come from two kinds of restoring. One is a body growing back what it lost: scientists are learning how a frog, a mouse, or a young child can regrow tissue that was cut away. The other is a country trying to repair a wrong it did to its own people. The numbers in this quiz come from both, and the last part of the Guide asks a question that runs through all of it: does a country have to choose?

Slope: how much it goes up for each step across

Picture a tissue growing back after an injury, and suppose (these are made-up numbers, to learn with) that it is 2 millimeters long at the start and grows a quarter of a millimeter every day. After 4 days it is 3 millimeters long. After 8 days it is 4 millimeters. Plotted on a grid, with days across and length up, those points lie on a straight line.

The slope of the line is how much it goes up for each step it goes across. From day 4 to day 8, the line goes across 4 days and up 1 millimeter, so the slope is 1 divided by 4, which is 0.25 millimeters per day. Up is called the rise; across is called the run:

slope = riserun

When you have two points instead of a picture, you find the rise and the run by subtracting. Take the two points (4, 3) and (8, 4). The rise is the change in the second numbers, 4 − 3 = 1. The run is the change in the first numbers, 8 − 4 = 4. The formula sheet you get on the test writes this as

m = y₂ − y₁x₂ − x₁

The letter m is the usual name for slope. The little 1 and 2 only mean “the first point” and “the second point.” The one rule is to take them in the same order on the top and the bottom. If you start with the second point on top, start with the second point on the bottom too.

The sign of the slope tells you which way the line goes. Reading from left to right:

• A line that goes up has a positive slope.
• A line that goes down has a negative slope.
• A flat line has a slope of zero: it goes across and never up.
• A line that goes straight up and down has no slope at all. Its run is zero, and you cannot divide by zero, so its slope is called undefined.

Slope is a rate

In a word problem, slope is almost always a rate: so much of one thing per one of another. Its unit is the unit going up, per the unit going across. In the tissue example it is millimeters per day.

Here is one with real numbers. In 1988 the United States passed the Civil Liberties Act. It apologized for imprisoning about 120,000 Japanese Americans in camps during World War II, and it paid each person who was still living $20,000. The total paid is a straight line: 1 person, $20,000; 10 people, $200,000; 50 people, $1,000,000. Take two of those points, (10, 200,000) and (50, 1,000,000). The rise is 1,000,000 − 200,000 = 800,000 dollars. The run is 50 − 10 = 40 people. The slope is 800,000 ÷ 40 = 20,000 dollars per person, which is exactly the payment. In a word problem, the slope is the rate the story gives you: here, $20,000 for each person.

Where a line crosses the axes

A line usually crosses both axes, and each crossing point has a name and a meaning.

• The y-intercept is where the line crosses the up-and-down axis. There, x is 0, so it is the starting value: the amount before anything has happened.
• The x-intercept is where the line crosses the across axis. There, y is 0, so it is where the amount runs out, or where it began from nothing.

A real example. In 2019 the city of Evanston, Illinois, set aside the first $10 million of a new tax on cannabis sales to pay reparations to Black residents for housing discrimination in the city between 1919 and 1969. Each grant was up to $25,000. If every grant is the full $25,000, the money left after g grants is

y = 10,000,000 − 25,000g

The y-intercept is $10,000,000: before any grant, the whole fund is there. For the x-intercept, set y to 0 and solve: 0 = 10,000,000 − 25,000g, so 25,000g = 10,000,000, and g = 400. After 400 grants the money is gone. The slope is −25,000 dollars per grant, negative because the line goes down: every grant takes $25,000 out.

The equation of a line: y = mx + b

Every straight line that is not straight up and down can be written as

y = mx + b

where m is the slope and b is the y-intercept. This is called slope-intercept form, and it is on the formula sheet. Read it as a sentence: start at b, and add m for every step across. The tissue line starts at 2 and adds 0.25 a day, so it is y = 0.25x + 2. The fund starts at 10,000,000 and loses 25,000 a grant, so it is y = −25,000x + 10,000,000, the same equation as before with the parts in a different order. The Civil Liberties Act line starts at 0, because no people means no money paid, so it is y = 20,000x + 0, or just y = 20,000x. A line with b = 0 goes through the corner of the grid, the point (0, 0), and that is what a proportional relationship looks like.

When you know the slope and one point, but not b. The formula sheet has a second form for this, called point-slope form:

y − y₁ = m(x − x₁)

Put in the slope and the point, then solve for y. Say the slope is 3 and the line goes through (2, 5). Then y − 5 = 3(x − 2). Multiply out the parentheses: y − 5 = 3x − 6. Add 5 to both sides: y = 3x − 1. Check that (2, 5) is on it: 3 × 2 − 1 = 5. It is.

When the equation comes in another shape. Some questions give a line as something like 2x + y = 7. That is called standard form, with both variables on one side. To read its slope, solve for y: take 2x from both sides, and y = −2x + 7. Now it says slope −2, y-intercept 7. If there is a number in front of y, divide by it at the end: 4x + 2y = 10 becomes 2y = −4x + 10, then y = −2x + 5.

Flat lines and upright lines

A line whose value never changes is flat. A single payment of $25,000, counted year after year, is $25,000 in year 1 and $25,000 in year 10. Its slope is 0, so its equation is y = 0x + 25,000, or just y = 25,000. Every flat line has the equation y = a number.

A line that goes straight up and down is the opposite: every point on it has the same x. The line through (4, 0), (4, 1) and (4, 7) is x = 4. Its slope is undefined, and it is the one kind of line that cannot be written as y = mx + b.

Parallel and perpendicular

Parallel lines never meet. They go the same direction at the same steepness, so they have the same slope and different y-intercepts. y = 2x + 1 and y = 2x − 4 are parallel.

Here is a pair of parallel lines that matters for this quiz. Suppose a country has 100 units of money to divide between two things, say space exploration and ending poverty. If x goes to space and y to poverty, then x + y = 100. Solved for y, that is y = −x + 100: a line with slope −1. Every unit more for space is a unit less for poverty. That is what people mean when they call a choice zero-sum: one side can gain only what the other side loses.

Now suppose the country has 120 units instead. The line is x + y = 120, or y = −x + 120. It has the same slope, −1, so it is parallel to the first line, but it sits farther out. Spend 40 on space, and the first line leaves 60 for poverty while the second leaves 80. The trade-off between the two has not changed. What changed is the total. When the total grows, both sides can have more.

Perpendicular lines cross at a square corner, a right angle. Their slopes are flipped and switched in sign: if one slope is 2, the other is −12. If one is −34, the other is 43. A quick check is that the two slopes multiply to −1: 2 × (−12) = −1.

Two lines at once: a system

A system of equations is two equations that are both true at the same time. On a graph, each one is a line, and the answer is the point where the two lines meet. That point is the only pair of numbers that sits on both lines.

Here is one to picture. Suppose (again, made-up numbers) that a person who is owed a repayment is offered two choices. Plan A pays $25,000 once. Plan B pays $5,000 now and $2,500 at the end of every year. After t years, in thousands of dollars, plan A has paid y = 25, and plan B has paid y = 5 + 2.5t. When has plan B caught up?

On the graph the lines meet at (8, 25): after 8 years, each plan has paid $25,000. After that, plan B has paid more. You do not need a graph to find this, and on the test you usually will not have one. There are two ways to do it with the equations.

Substitution: put one equation into the other. Plan A says y is 25. So in plan B’s equation, put 25 in place of y: 25 = 5 + 2.5t. Now it is a one-variable equation, solved the way Quiz 33 solved them. Take 5 from both sides: 20 = 2.5t. Divide both sides by 2.5: t = 8. Then find y from either equation: y = 25. The answer is (8, 25), the same point the graph showed.

Substitution works whenever one equation already says what one variable equals. If y = 3x + 1 and y = −x + 9, both right-hand sides equal y, so they equal each other: 3x + 1 = −x + 9. Add x to both sides and take 1 from both: 4x = 8, so x = 2, and y = 3 × 2 + 1 = 7.

Elimination: add or subtract the equations to get rid of one variable. This works best when both equations come in standard form. Suppose (made-up numbers again) a city gave out 30 grants, some of them $25,000 housing grants and the rest $10,000 repair grants, for a total of $600,000. With h housing grants and r repair grants, and the money counted in thousands:

h + r = 30
25h + 10r = 600

Multiply every term of the first equation by 10, so that both equations have 10r in them: 10h + 10r = 300. Now subtract that from the second equation. The 10r terms cancel, which is the point: 25h − 10h = 600 − 300, so 15h = 300 and h = 20. Then r = 30 − 20 = 10. Check both equations: 20 + 10 = 30, and 25 × 20 + 10 × 10 = 500 + 100 = 600. Twenty housing grants and ten repair grants.

Two things can happen that give no single answer. If the two lines are parallel, they never meet, and the system has no solution: same slope, different intercept. If the two equations turn out to be the same line written two ways, every point on it works, and there are infinitely many solutions. The test sometimes asks which of these it is, and the slopes and intercepts tell you.

Does a country have to choose?

In July 1969, the day before Apollo 11 left for the moon, a civil rights leader named Ralph Abernathy brought a protest to the launch site in Florida, with farm wagons pulled by mules. He asked why the country could send men to the moon and not feed its own poor. The head of NASA met him at the gate. The interlude at the end of this quiz tells what they said to each other.

The argument between them is the budget line above. In any one year, with the total fixed, the line is real: money spent in one place is not spent in another. But the total is not fixed from one decade to the next. A country’s wealth can grow, and when it does, the line moves out, parallel to where it was. Whether a choice is zero-sum depends on whether the total can grow. That is a question the math can state clearly. It cannot answer it for you.

Three to study before you start

Now you

Work on paper. Use the formula sheet if you like; the slope formula and both forms of a line are on it. Questions 4 and 9 are multiple choice.

1. Find the slope of the line through (2, 5) and (6, 17).
2. For the line in this graph, find (a) the slope, (b) the y-intercept, (c) the equation, and (d) the x-intercept.
3. A model of a regrowing tissue, with made-up numbers. Find the slope and say what it means, write the equation, and find the length at week 10.

   week	0	2	4	6
   length (mm)	3.0	4.2	5.4	6.6
4. The money left in a reparations fund after g grants is y = 10,000,000 − 25,000g. What does the x-intercept of this line tell you?
   • A) The fund starts with $10,000,000.
   • B) Each grant is $25,000.
   • C) After 400 grants, the money is gone.
   • D) The fund grows by $25,000 with each grant.
5. A line has slope −3 and goes through the point (2, 7). Write its equation in the form y = mx + b.
6. The line 4x + 2y = 10. (a) Write it as y = mx + b. (b) Write the equation of the line parallel to it that goes through (0, 1). (c) What slope would a line perpendicular to it have?

Reading A reading rest stop, for the stubborn ones.

Read While NASA Was Landing on the Moon, Many African Americans Sought Economic Justice Instead, by Bryan Greene, in Smithsonian magazine. Land on the part about Ralph Abernathy’s march to the launch site, and find what the head of NASA said about pushing the button. Then come back for the last four questions.

The photograph above is NASA’s: the road to the Kennedy Space Center before dawn on July 16, 1969, lined with people waiting for the launch. It is not from the article.

7. Plan A pays $25,000 once. (a) If y is the total it has paid after t years, write its equation and give its slope. (b) What is the equation of the straight up-and-down line through the point (4, 3), and what is its slope?
8. Solve the system by substitution: y = 3x + 1 and y = −2x + 16.
9. Plan E pays $6,000 now and $2,000 a year. Plan F pays $2,000 now and $3,000 a year. After how many years have they paid the same amount, and what is that amount?
   • A) 2 years, $10,000
   • B) 4 years, $14,000
   • C) 5 years, $17,000
   • D) Never: the lines are parallel.
10. A city gave out 40 grants, some for $25,000 and the rest for $10,000, for a total of $775,000. How many of each? Write the two equations, then solve by elimination.

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