#word problem only using linear equation

115 messages · Page 1 of 1 (latest)

dapper aurora
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Two classmates poorly copy the text of an exercise written on the blackboard by the teacher. The exercise consists in determining the point of intersection of the line of equation y = 2x with another line, of which the teacher provides the equation. The teacher also provides as a solution 3/4 3/2 The second straight line has an equation of the type y = mx + g, m and q being two integers. Luke makes a mistake in transcribing m and Mark makes a mistake in transcribing q. Luca gets the solution (-3, -6) while Marco gets the solution (1, 2). Both correctly solved the exercise that they had mistakenly transcribed. Would you be able to determine the equation of the second line assigned by the teacher and the equations erroneously transcribed by Luke and Mark, knowing that the original value of q given by the teacher was a positive integer equal to and not exceeding 10?

still sparrowBOT
lofty kernel
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You know that mx+q passes throught 3/4, 3/2 so you can get an equation in m and q

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Let's call the line that Luke/Luca writes y=lx+q and the line that Mark/Marco writes y=mx+a

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Then you can also get equations in l, q and in m, a using the points they get as answers

lofty kernel
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Answer:||No, you can't determine those things. There are in fact three possibilities||

dapper aurora
lofty kernel
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What do you think they are?

dapper aurora
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3 equations

lofty kernel
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But what are the possible second lines?

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That's my question for you

dapper aurora
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@lofty kernel can you continue?

lofty kernel
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You get 3 equations, can you please send what you get

dapper aurora
lofty kernel
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Where does it say q is even?

dapper aurora
lofty kernel
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Maybe I am blind but I don't see the word even

dapper aurora
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Oh

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I didn’t put it in then

lofty kernel
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Oh okay then btw there is only 1 possibility for the second line

lofty kernel
dapper aurora
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The original problem is in a another language

dapper aurora
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But I can’t think

lofty kernel
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So original second line is y=mx+q, Luke writes down y=lx+q and Mark writes down y=mx+a

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You understand that right

dapper aurora
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Yeah

lofty kernel
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Now you know the solutions they all get

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So you can plug those points in the equatiin

dapper aurora
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X and y values?

lofty kernel
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Yes

dapper aurora
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But I’d be missing m and q to solve them

lofty kernel
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Yes but just substitutie the x and y values and send me what you get

dapper aurora
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Alright

lofty kernel
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Then we will solve for m and q

dapper aurora
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m(-3)

lofty kernel
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You should get
$\frac{3}{2}=\frac{3}{4}m+q$\
$-6=-3l+q$\
$2=a+q$

hoary otterBOT
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Ryanstaal

713803803291287654.png
dapper aurora
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I don’t understand

lofty kernel
dapper aurora
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Is that 1 equation?

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Oh nvm

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I see

lofty kernel
dapper aurora
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Why no x value in the third one

lofty kernel
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Your lines were kind of correct but you dient use the different letters

lofty kernel
dapper aurora
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Oh

lofty kernel
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And 1*a=a

dapper aurora
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Right

lofty kernel
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Now look at the second equation -6=-3l+q

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That tells you that q is divisible by a certain number

dapper aurora
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Why?

lofty kernel
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q=3l-6

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You can kind of factor the right hand side to get that q is divisible by a certain number

dapper aurora
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Y

lofty kernel
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What?

dapper aurora
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Yes

lofty kernel
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So which number divides q? (Except 2)

dapper aurora
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6?

lofty kernel
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Yes

dapper aurora
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I don’t get why divide though

lofty kernel
dapper aurora
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Why would you divide that

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Is there a rule

lofty kernel
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q=3l-6=3(l-2)

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But that's basically the definition of divisibility by 3

dapper aurora
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Could you use the same letters?

lofty kernel
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Because q/3=l-2 and right hand side is an integer

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So left hand side is integer too

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So q is divisible by 3

dapper aurora
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Yes

lofty kernel
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So we have now that q is divisible by 2 and 3

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But we also know that q is a positive integer less than or equal to 10

dapper aurora
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It has to be 6 then

lofty kernel
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Exactly

lofty kernel
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Then solve for m, l, a and you have the equations of all your lines

dapper aurora
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Wait

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It’s right

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Hold on

dapper aurora
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Why divide

lofty kernel
dapper aurora
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I’m not getting the division thing

lofty kernel
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So I hoped that would make it clearer

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If you don't understand you can also just brute force by plugging in q=2, 4, 6, 8, 10 and see what values you get for the other variables

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You will see that you won't get integer values for l

dapper aurora
lofty kernel
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It's just ab+ac=a(b+c)

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What grade are you in btw?

dapper aurora
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10

dapper aurora
lofty kernel
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??

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It's just factoring something out

dapper aurora
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Oh

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Welp I still don’t get that, it’s probably just me

lofty kernel
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Maybe someone else can explain it better <@&286206848099549185>

dapper aurora
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Wait how could I forget

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It’s just an equation

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To find l

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Or not

lofty kernel
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That's not how I would describe it

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But the important thing is that you understand how to get that q is divisible by 3

dapper aurora
lofty kernel
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Yeah I think you should just try 2, 4, 6, 8 and 1 for q

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And then solve for l

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And what value(s) of q will give you an integer for l

dapper aurora
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Then for the second one just get the m and q values from the second and third equation