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Common Core Standards: Math

High School: Algebra

Reasoning with Equations and Inequalities HSA-REI.D.10

10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Students should understand that equations with two variables can be represented graphically. The shape that results on the coordinate plane is a visual representation of all the solutions to that equation.

What does that mean? It means we're not just pulling rabbits out of hats! The equations actually mean something visually.

An equation with two variables can be anything from y = x to x2 + y2 = 4 to 19x13 = y. Some are simpler than others, of course, but they all have an x and a y.

That means instead of having an equation with one variable (and therefore one solution), we can have many different solutions. Graphically, we can represent these solutions by drawing a curve or line through all the pairs of solutions (one for x and one for y) that work for that particular equation.

Let's take the equation 7x – 18 = y and see how we can represent this graphically.

How do we prove to a student that indeed, a line of a two-variable equation, when graphed, shows all of the solutions? Let's show them how to pull that rabbit out of the hat themselves.

Since any two points define a line, all we need to do is input two values for x and see what the output y values are. We'll pick three points just to be sure our graph is a line and not some weird curve.

Let's pick the numbers -1, 0, and 3 for x. Plugging in the numbers for x into our equation 7x – 18 = y gives us -25, -18, and 3 for the y values. So the points in our graph become (-1, -25), (0, -18), and (3, 3). If we graph these on the x-y coordinate plane, we'll have this:

If your students don't believe you, prove to them that the equation and graph correspond to one another. Take a point on the line that is easily identifiable, say (2, -4), and plug the values into the equation. If we do that, we'll have -4 = 7(2) – 18, which simplifies to -4 = -4. That way, students will be sure that points on the line or curve are valid solutions to the equation, and vice versa. 

But don't stop there. It's also important to prove the opposite. For example, the coordinate (4, 1), which is not on the line, is also not a solution to our equation. If we plug in the coordinates, we can confirm this: 1 = 7(4) – 18 is false because 1 ≠ 10. This means (4, 1) isn't a solution to our equation and not a point on the line.

Now you can pat yourself on the back and prove to students that teachers aren't just up to some magic tricks. Everything in math pretty much works as it's supposed to.

This method can be applied to two-variable equations of higher orders. The generic shapes of these equations (such as quadratics making a parabola) should be known and associated with each other already. Otherwise, students will need to graph several points before verifying the graph that corresponds with the particular equation.

Drills

  1. Dr. Frankenstein thinks he knows more than you about what is true and false world just because he's a doctor. (Just because he brought a corpse back to life, he thinks he's hot stuff.) He says that the equation y = 17x + 1 also includes the point (1, 8). Is Dr. Frankenstein right or wrong?

    Correct Answer:

    He's wrong

    Answer Explanation:

    By plugging our point into the equation, we get 8 = 17(1) + 1, which simplifies to 8 = 18. This is obviously false. Because we know our two-variable equation corresponds to a line on the graph (after all, it's in y = mx + b form), we don't even have to graph it to know that the point does not lie on the line.


  2. The Kooky Dough Company makes cookie dough, but it takes a little time for it to start reeling in the dough. The equation y = 2x – 8 models the profits y after making x pounds of cookie dough. What are the x and y coordinates of their break-even point?

    Correct Answer:

    (4, 0)

    Answer Explanation:

    The break-even point is when the profits are $0 exactly. Since y represents the profits, we can set it to zero and solve for x. If we do so, we have 0 = 2x – 8, which simplifies to x = 4. That means our answer is (4, 0). We can see this by plotting the graph, too.


  3. You talk on the phone y minutes on day x of every month according to the equation y = 2x + 1. The cell phone company claims you talked 12 minutes on the phone on the fourth day of the month. Are they right?

    Correct Answer:

    No, you talked on the phone for 9 minutes on the fourth of the month

    Answer Explanation:

    First, we can solve this mathematically by substituting in our variables, x = 4 and y = 12. When we compute 12 = 2(4) +1, we get 12 = 9, which is untrue. That means (A) isn't right. Instead, you talked 9 minutes (since we already found out that when we plug in x = 4 into the equation, our answer for y should be 9). We can see that the graph will say the exact same thing.


  4. The speed of a snowboarder changes from uphill to downhill at a speed of y = x2 + 1 where x is in minutes. The snowboarder's speed at time 0 is 1 and is 2 at time 1. The snowboarder claims that this proves his speed increases linearly. Is he right?

    Correct Answer:

    No, because the equation is not linear

    Answer Explanation:

    While (0, 1) and (1, 2) may seem like they make a straight line at first, we're given the equation y = x2 + 1. If we plug even one more point (say, x = 2), we'll get a point (y = 5) that doesn't work with our hypothetical line. This is because the equation isn't linear; it's quadratic. And just in case we want to really make sure, we can graph it, too.


  5. A financial model has been constructed to predict the total profits of a company within the next month. The relationship is given as y = x4 – 3, where y is the amount of profit and x is the number of days. When will the company start to gain money (as in y is greater than 0)?

    Correct Answer:

    Between day 1-2

    Answer Explanation:

    We can set y = x4 – 3 to equal zero and then find x. If we do that, we'll end up with , which gives x ≈ ±1.32. That means the company will make 0 profits after 1.32 days (since negative days can't exist), but we don't know whether they'll start to make money or lose money. If we calculate y for x = 2, we'll have y = 24 – 3 = 13. When x = 1, the y value is y = 14 – 3 = -2. Since the profits go from negative to positive, the right answer is (B).


  6. The rate at which humans breathe oxygen aboard the International Space Station is pretty steady. The equation y = x – 12 shows this to be the case, where x represents the number of people and y represents the relative danger of the oxygen supply. What is the maximum number of people allowed aboard the space station before the oxygen supply becomes dangerous (y > 0)?

    Correct Answer:

    12

    Answer Explanation:

    If we find how many people it takes for y to equal 0, we will know how many people it takes for the oxygen supply to reach dangerous levels. A quick look at the equation will tell us that y = 0 when x = 10. Since the oxygen levels are dangerous only when y is greater than zero (not greater than or equal to), 10 is the maximum number of people that can be on the international space station. We can confirm this by graphing the equation.


  7. Does the point (2, -3) exist on the line y = x?

    Correct Answer:

    No, point (2, -3) is not on the line

    Answer Explanation:

    The perfect way to test this is to plug in the coordinates of the point into the given equation. If x = 2 and y = -3, let's substitute those numbers for y = -x. That would give us -3 = -2, which is false. Therefore, the point (2, -3) is not on the line y = -x.


  8. If a graph for our equation shows that the line doesn't pass through a certain point than we can assume that:

    Correct Answer:

    The equation, when solved for that point, will be false

    Answer Explanation:

    We have to understand that the equation and the graph of the equation are inseparable! They are different (written and visual) representations of the same idea. Hence, if one is false, the other one will be false as well.


  9. Which of the following is a two-variable equation?

    Correct Answer:

    Both (A) and (C)

    Answer Explanation:

    Both (A) and (C) have two variables that must be solved for (we could call them x and y, but why put labels on everything?), whereas (B) only has one.


  10. Which of the following two-variable equations, when plotted on a graph, would be a curve (rather than a line)?

    Correct Answer:

    y = x2

    Answer Explanation:

    You should automatically see that any variable that is raised to a certain power will not be a straight line.


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