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Common Core Standards: Math

High School: Algebra

Reasoning with Equations and Inequalities HSA-REI.C.6

6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

By now, students should know that a linear equation given by y = mx + b (or a variation of it) and makes a line on a graph. They should also know that a system of linear equations means that we have more than one y = mx + b equation in the mix. So far, so good.

Solving a system of linear equations means finding the point at which the two (or more?) lines intersect. This happens when the same set of x and y values satisfy all the linear equations in the system.

What about lines that never intersect? Well, they're parallel, for starters. But that just means that the system of parallel lines will have no solution. (It may be helpful to tell your students that "No solution" is a legitimate answer, but only after they've done the work to prove it. Otherwise you'll get "No solution" as the answer to every homework problem.)

Students should be shown that a system of linear equations can be solved either through graphs or straight algebra, but that these two methods arrive at the same answer because they mean the same thing. The goal is to find the point at which the two lines intersect.

Graphs

Graphically, solving a system of two linear equations is easy. Using the slopes and y-intercepts of both lines, students can graph each line individually and find the intersection point. It's pretty and simple (and pretty simple), but not always accurate visually. For instance, what's the solution to the following system of equations?

Is it at point (6.2, 17)? Or maybe (6.3, 17.5)? It's difficult to tell exactly. Very helpful visually, but a bit less useful in terms of coming up with numerical answers.

 

Algebra

It'd be helpful to know that the two lines above have equations of y = 2x + 5 and y = 3x – 1.25. With the linear equations, students should be able to graph the lines on a coordinate plane as well as solve them exactly using substitution. When the lines intersect, the x and y values in both linear equations are the same. That means we can set the two equations equal to each other.

For instance, we can say that 3x – 1.25 = 2x + 5 because both the y coordinates have to be equal. (By the way, that gives x = 6.25 as our answer. If we substitute 6.25 for x back into either equation, we should get 17.5.)

If students are struggling with these concepts, emphasizing the connection between the visual graphs and the algebraic calculations should help cement their understanding.

Drills

  1. As the CEO of a company, you need to figure out how quickly manufacturing costs are adding up. Your assistant tells you that the relationship of cost (y) to the number of products sold (x) is y = 3x + 13. The revenue, on the other hand, is y = 5x – 39. Assuming each product that is manufactured is also sold, how many units must you produce in order to start making money (meaning revenues are greater than costs)?

    Correct Answer:

    27

    Answer Explanation:

    We must find the point at which revenue are greater than costs. By substituting the first equation into the second one we get 3x + 13 = 5x – 39. If we simplify for x, we eventually get x = 26. This is our break-even point, where the cost equals the revenue. We must produce 1 more unit in order to make a profit. That means our answer is 27 units.

    In case you were curious, the graph looks like this.


  2. The space shuttle uses up fuel at a cost of y = 600x + 1500, where x is the number of gallons of fuel and y is our total cost. Boeing has decided to design a much more fuel-efficient rocket engine where the cost is calculated at y = 500x + 2700. Is this new engine actually more fuel efficient in the long run than the old one or not?

    Correct Answer:

    Yes, it is more fuel-efficient when x > 12

    Answer Explanation:

    While (C) is correct, we asked if the new engine is more fuel-efficient in the long run. Clearly, after 12 gallons of consumption, the new Boeing engine will be much more cost-effective. We can see this by finding the intersection of the lines. Setting them equal to each other, we have 600x + 1500 = 500x + 2700, which simplifies to x = 12. At x = 13, however, the new engine will cost 500 × 13 + 2700 = $9200, while the old one will cost 600 × 13 + 1500 = $9300.


  3. One salsa (the food, not the music style) factory produces y gallons of salsa for every hour the machine is running, given by x, at a rate of y = -30x + 900. (The machines get clogged up after a while so it makes less after a certain amount of time.) A second type of salsa at the same plant makes y gallons for every x hour at a rate of y = 15x + 90. When will the amount of salsa manufactured be the same?

    Correct Answer:

    After 18 hours

    Answer Explanation:

    If we want the number of gallons of salsa produced at that hour to be the same, we can set the two equations equal to each other. That means -30x + 900 = 15x + 90, which simplifies to x = 18. Both machines will produce salsa at the same rate y after they've each been working for 18 hours. The graph looks like this.


  4. You're skiing down a slope at a speed of y = 720x + 19 after x minutes. (Did we mention that you have a really cool jetpack?) Your friend's speed is calculated to be y = 83x + 89. (There was only one jetpack and you called dibs.) If you're going down the same slope in the same conditions, at which point in time will the two of you be traveling at the same speed?

    Correct Answer:

    0.11 min

    Answer Explanation:

    The speeds (or the y values) should be equal at a certain time (or x value). To find the time, we can set the speeds equal to each other, creating the equation 720x + 19 = 83x + 89. Simplified, we get x = 0.11 minutes. This answer makes sense in the context of the problem, since your jetpack propels you to go way faster than your friend.


  5. An Olympic swimmer is able to cover a distance y after x minutes of swimming, given by y = 300x. You aren't as fast of a swimmer (you're better at math, though, so it's okay). You cover y meters as y = 100x + 300 where x is in minutes after the start of the race. At which point in time will the Olympic swimmer catch up to you?

    Correct Answer:

    1.5 minutes

    Answer Explanation:

    The Olympic swimmer will catch up to you after 1.5 minutes since they are faster (300x vs. 100x), but you had a head start of 300 meters. We can find this by setting the equations equal to each other like so: 300x = 100x + 300. This simplifies to x = 0.5.


  6. A racer only puts in so much fuel into his tank since the more he puts in, the higher the energy expenditure needed to drive the car. He will have y = -40x + 1200 gallons of fuel left in his tank after x seconds after stepping onto the gas. His opponent will have y = -2x + 600 gallons of fuel left. After how many seconds of full throttle racing will they have the same amount of fuel left in the tank (rounded to the nearest second)?

    Correct Answer:

    16

    Answer Explanation:

    After putting the petal to the metal, we can set the two linear equations equal to each other and get -40x + 1200 = -2x + 600. Once we simplify this, we should end up with 15.79 = x. Since we're asked for the answer to the nearest second, that would be (A).


  7. The Delicious Donut Shop knows that it can sell y = 32x doughnut in x hours. Their competitors, Donutello's, can sell y = 32x + 3 doughnuts after x hours. After how many hours will the Delicious Donut Shop catch up?

    Correct Answer:

    Never, the lines are parallel

    Answer Explanation:

    We don't even have to mathematically solve it or graph it. The slope is the same for both lines, which means they are parallel. They will never intersect, and Donutello's will always sell 3 more donuts than the Delicious Donut Shop at any given hour.


  8. At which point in time will two lines with the same slope intersect one another?

    Correct Answer:

    Never

    Answer Explanation:

    If you have two linear equations you must solve for, and they both have the same slope, they are parallel, and will not intersect. Simple as that.


  9. A linear equation, when plotted will look like a:

    Correct Answer:

    Straight line

    Answer Explanation:

    A linear equation by definition is itself a straight line. And that's all it'll ever be.


  10. To graphically solve a system of linear equations you need to plot the graph and find the:

    Correct Answer:

    All of the above

    Answer Explanation:

    All of the answers mean the same thing, essentially! Did we trick you? Hopefully not.