Question

Graph the solution set of the system of inequalities or indicate that the system has no solution.

Answer 1

x^2 + y2^ ≤ 49
5x + 4y ≤ 20
@Psymon could you help me with this?

Answer 2

Hmm, a circle and a line, lol.

Answer 3

i am not good with graphing at all!!

Answer 4

Well, in this case our elimination method from before isn't much of an option. Probably impossible really. This means we must do substitution. The idea behind substitution is we pick one equation and then get either x or y in that equation all by itself. Just as if we were solving a normal algebra problem like 2x -5 + 13. We would get x by itself. Same thing with this.

Now for the top equation, getting x or y by itself isn't practical. It doesn't do any good. So we want to use the bottom equation and see if we can solve for x or y. Which one we solve for does not matter.

Since y has the smaller number multiplying it, I'll just solve for y in the second equation. So we have:
5x+4y <= 20

I first subtract 5x from both sides and then divide everything by 4

5x + 4y <= 20
-5x -5x

4y <= 20 -5x\[y=\frac{ 20-5x }{ 4 }\]

So now we've solved for y. This result we then must substitute intot he other equation. Make sense so far?

Answer 5

Yea i think i got it so far, its making sense!

Answer 6

Okay, awesome. So here comes probably the more difficult part. We have to take what we just came up with and replace it for y in the other equation. Meaning we have this:

\[x ^{2}+(\frac{ 20-5x }{ 4 })^{2}=49\]So yeah, pretty ugly here. But this is doable. Me personally, I think I would want to rewrite that fraction squared part. Be aware that the numerator is being squared and the denominator is being squared. What I'm going to do is ONLY square the bottom. Doing that will let me manipulate some things. So this is what I get:

\[x ^{2}+\frac{ (20-5x)^{2} }{ 4^{2} } =49\]
\[x ^{2}+\frac{ (20-5x)^{2} }{ 16 }=49\]

So now that I have this, I can eliminate the fraction. Because I have a division of 16, I can eliminate it by a multiplication of 16, the opposite. This requires me to multiply everything by 16. That will give me:

\[16x ^{2}+(20-5x)^{2}=(49)(16)\]
Notice how I didn;t actually multiply 49 and 16, I just put them next to each other like a multiplication. The reason I do this is sometimes it is not necessary to multiply right away. We might be able to cancel something or simplify things easier by not multiplying. Anyway, if I multiply them and get 784, I can't even tell there's a 16 or a 49 in there. It's a lot harder to see the factors.

Okay, so now I have to square that (20-5x) portion. There is a formula for a binomial squared that is:
\[(a-b)^{2}=a ^{2}-2ab+b ^{2}\]Now I'll still multiply it out the normal foil way to show you, though. So I have
(20-5x)(20-5x) So foil now

20*20 = 400
20*-5x = -100x
-5x*20 = -100x
-5x*-5x = 25x^2

combining them I get

25x^2 - 200x + 400. So now let's see what I have when I combine everything:

16x^2 + 25x^2 - 200x + 400=(49)(16)

41x^2 - 200x + 400-(49)(16) = 0

Now I move everything to one side of the equation because this is going to be a quadratic polynomial, meaning we have to factor it. Problem is the factoring of it looks like it'll be quite annoying. We can still try, though. Now, looks like I'm going tobe forced to multiply out that (49)(16), so I do that to get 784. SO 400-784 = -384, giving us

41x^2 - 200x - 384 = 0

Okay, this is ridiculous, haha. This is the process you'd have to do, though x_x I'm guessing the equation isn't expecting you to get the actual solutions, just get a general idea because I mean.......yeah, the answers for this are insane. But the idea is do you get what I did up to this point?

Answer 7

i got it so ar i probably wont remember everything but i got it so far.. This is a lot :0

Answer 8

Well, this isnt something that can easily be done by hand. Without a graphing calculator like I have, this requires calculus to get good results. But here's what we can do. We can still graph both of the equations and then get a good estimate of what our answers will be. Because these are inequalities, we don't have to be exact, we can be close and still get correct answers.

Answer 9

So, do you think you would know how to graph the bottom of the two equations?

Answer 10

i have know idea how to graph it .. Im sorry!! & I know i should have bout a graphing cal but i thought the one i bought had graphing but it doesnt and im half done with my pre calc so i thought i shouldnt get one.

Answer 11

I wouldnt get a graphing calculator unless youre going to docalculus and higher. Theres no point to it with anything less than that. But okay, do you remember

y = mx + b?

Answer 12

a little but not a lot though.

Answer 13

Do you remember what m and b mean in that equation by chance?

Answer 14

b is the y-intercept i believe and i think m is the slope?

Answer 15

RIght. Hence why it is called slope-intercept form. The random m and b in the equation represent slope and intercept. The idea is when we have x and y to the first power, we can make the equation look like

y = mx + b

When we make it look like that, it is very easy to graph it. So given we have
5x + 4y <= 20, we now solve for y. Now we already did solve for y way up above and we got:

\[y = \frac{ 20-5x }{ 4 }\] This isnt quite y = mx + b, but we only need a slight change. Now, every term in the numerator of a fraction can be made into its own separate fraction. For example:

\[\frac{ x ^{3}-x ^{2}-x - 1 }{ 5 } \]Here I have a fraction with 4 terms in the numerator. This means I can make 4 separate fractions. All I do is take one of those terms and put it on top of our original denominator. So when i split this into 4 fractions, i get:

\[\frac{ x ^{3} }{ 5 }-\frac{ x ^{2} }{ 5 }-\frac{ x }{ 5 }-\frac{ 1 }{ 5 } \]


So what I just did for this example I can do for our problem. So:

\[\frac{ 20-5x }{ 4 } \implies \frac{ 20 }{ 4 }-\frac{ 5x }{ 4 }\]

\[y = 5 - \frac{ 5x }{ 4 } \]
So now we have the form we want. Can you tell what the slope and what the y-intercept are?

Answer 16

is slope 5x? and y-intercept 4?

Answer 17

Nope. The y-intercept is the number not with x. So that means the y-intercept is +5. The slope is the number paired with x. Now the number with x in this case is NEGATIVE (5/4), We have to be wary of the sign, too. So what a -5/4 slope tells us is each point is down 5 and right 4 from the previous point. So we continuously go down 5 right 4 down 5 right 4, etc, etc. And our line crosses the y-axis at y = 5. So with that information, we can definitely graph this:



Kinda make sense?

Answer 18

kinda makes sense yea..

Answer 19

Alright, cool. So now we need to graph the circle. I dont suppose you have any idea about how to do that, do you?

Answer 20

nope i sure dont.. i feel bad i dont know how to do anything:/

Answer 21

Well, I still want to ask to be sure xD

Answer 22

i understand lol. if i knew it i would surely be happy to do it.

Answer 23

Okay, so the formula for a circle is this:

\[(x-h)^{2}+(y-k)^{2}=r ^{2} \]Now in terms of remembering the formula, if you know pythagorean theorem, then you know:

\[a ^{2}+b ^{2}=c ^{2} \]Well, thats exactly what a circle is! The only difference is the -h and the -k in there, otherwise its pretyt much a^2 + b^2 = c^2. In fact, it even means the same thing



The -h and -k parts indicate where the center of the circle is. The h is the x-coordinate and the kis the y-coordinate. Now we need to be careful. If I have:
\[(x-2)^{2}+y ^{2}=16 \], this DOES NOT mean that h = -2 andour x-coordinate is -2. Even though it says minus, it's subtracting a positive number. The x-coordinate is actually positive 2. I

If I have:

\[x ^{2}+(y+3)^{2}=28 \]Now this time we see +3 but again, this does not mean that the k is a positive 3. Its actually -3. The reason is remember that I posted the formula with -h and -k. These have to be negative. But in this example I had a +3. Well, the only way thats possible is if its (y-(-3))^2 = (y+3). This is why the y-coordinate of the center of the circle is -3. Now also notice that your problem is just x^2 + y^2 = 49. There is no (x-5)^2 or anything like that. This means the center of the circle is the origin.

A long explanation, especially since your circle is at the center, but we still need to know what to do if we got a problem that had this -h -k thing. Dont know if any of that made sense xD