Interpreting linear equations in different forms
At about 264-271s the narrator says granola bars decrease by one for each extra smoothie, then corrects it to two fewer (slope -2). The interventions use the corrected value.
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Elena sells smoothies for $4 and granola bars for $2 and earns $60, which is written in standard form as 4x + 2y = 60. The video graphs it using the intercepts (0, 30) and (15, 0) and explains that any point on the line is a combination earning $60. It then rearranges the equation into slope-intercept form, y = -2x + 30, and interprets the slope: one more smoothie means two fewer granola bars.
You'll be able to
- write a standard form equation from a real situation.
- graph a standard form equation using its x- and y-intercepts.
- rearrange standard form into slope-intercept form and interpret the slope and intercept.
Key words
- standard form
- A linear equation written as Ax + By = C.
- slope-intercept form
- A linear equation written as y = mx + b, where m is the slope and b is the y-intercept.
- intercept
- Where a line crosses an axis: the x-intercept has y = 0 and the y-intercept has x = 0.
- slope
- How much y changes when x increases by 1.
Transcript
- [Instructor] We're told that Elena sells smoothies for $4 each, and granola bars for $2 each at the farmer's market. By the end of the day, she earns $60. So what I wanna do is I wanna express this statement in an equation, and then I wanna see if I can graph that equation, and then maybe I'll manipulate it algebraically different ways to see what I can tell about it.
So let's say that x is equal to the number of smoothies. Number of smoothies. She smell.. I was gonna say, she smells. She sells, and y is equal to the number of granola bars. Granola bars. So what would be an equation that would represent all of this information?
Well, the amount of money that Elena gets from smoothies is going to be, she sells smoothies for $4 each. So $4 times the number of smoothies. That's how much Elena gets from selling smoothies. And then how much does she get from granola bars? Well, the number of granola bars times $2.
So it's gonna be $2 times y. So this is gonna be the total amount of money that she makes. The left is from smoothies. This is from granola bars. And the total is $60. And what I've just done here is I've written an equation in standard form. You might recognize standard form.
Some people would say Ax plus By is equal to C. One of the most useful things about standard form, one, it's easy to translate between what I just wrote and the statement up here, or vice versa. And it's also pretty straightforward to graph it by looking at the x and y intercepts.
For example, I can see that when x equals 0, so if x equals 0, we're going to get 4 times 0 plus 2 y is equal to 60. We would get y is equal to, y is equal to 30. So we know that the point 0, 30 is on this line someplace. And then we could go the other way. What happens when y equals 0?
When y equals 0 we get 4x is equal to 60, x is equal to 15. So we know when y is equal to 0, x is equal to 15. And so if I get some graph paper out, I can graph these two points and connect them. So 0, 30... So 0, 30, it would be right over in here. That's this point, 0, 30.
And then 15, 0 would be, 15, 0 is right over there. And so then if I were to connect these two points, I'm not gonna do it, I'm hand drawing it, so it's not going to be perfect, but eh, not too bad. This right over here is the line of 4x plus 2y equals 60. Now how would I interpret this?
Well, what this means is any point on this line, and we have to be careful here because I'm assuming Elena can't sell parts of smoothies or parts of granola bars. Actually, let's assume that she can, she can sell parts of smoothies and parts of granola bars. Well, if she can, well then any point on this line could be a possible scenario where it would add up to, she would make $60 total.
And I'll try to find one that looks a little bit more... So maybe that point on that line, it would represent some number of granola bars and some number of smoothies so that she is making a total of $60. I haven't drawn this precisely, but hopefully you get that idea.
Now there's other ways of manipulating this equation or representing this equation. For example, I can start with 4x plus 2y. Let me rewrite it. 4x plus 2y is equal to 60. I could solve for y. So I could subtract 4x from both sides. Subtract 4x from both sides.
And I would get 2y is equal to negative 4x plus 60. I could divide both sides by 2 and I would get, lemme put it over here. I would get y is equal to negative 2x plus 30. Now you might recognize this as slope intercept form. Notice when x equals 0, y is equal to 30.
That is the y intercept. And then our slope over here, you might recognize, is negative 2. Every time we increase by one in the smoothies direction, in the x direction, we are going to decrease by one in the granola bars direction, in the y direction. So if we have one more smoothie, that means we are selling if we...
Yes, that's right. If we have one more smoothie, that means we have to sell two fewer granola bars. And so on and so forth. So that's another way of interpreting it. Some people actually find this method easier to understand. Some people find that method, but it's usually depending on the scenario that you're looking at.
They're still representing the exact same thing. They are just algebraically manipulated to look a little bit different.
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