Question

Please help me solve the following problem. Where does the line x=1+3t , y=2-t intersect the parabola y=x^2?

Answer 1

first step is to convert the parametric form of the line into the standard form involving just x and y. i.e. eliminate 't' from the equations.

Answer 2

equation of the line:
\[y=-\frac{ 1 }{ 3 }x+\frac{ 7 }{ 3 }\]

Answer 3

good - next step is to substitute this into the equation of the parabola - you will end up with a quadratic involving x.

Answer 4

ok so...

\[x ^{2}=-\frac{ 1 }{ 3 }x+\frac{ 7 }{3 }\]

Answer 5

and then just solve using the quadratic equation?

Answer 6

yes - that is correct - multiply through by 3 to remove the fractions and then solve.

Answer 7

\[3x^2+x-7=0\]

Answer 8

yup - solve to get two values for x, then use these to find the corresponding to values for y from the equation of the line you got above.

Answer 9

or even simpler, use \(y=x^2\) to find the corresponding values for y once you have the values for x.

Answer 10

\[y=(\frac{ -1\pm \sqrt{85} }{ 6 })^{2}\]

Answer 11

perfect!

you can see a pictorial representation of this solution here: http://www.wolframalpha.com/input/?i=intersection+of+y%3Dx^2+and+3y%2Bx%3D7

Answer 12

Ok thanks and how do I get an exact answer from the equation I just posted? How do I square the numerator?

Answer 13

oh wait nvm i think i know, gimme a sec

Answer 14

just use a calculator :)

Answer 15

BTW: what you typed IS the EXACT answer. Showing it as a decimal is an approximation.

Answer 16

But can't what I typed be simplified?

Answer 17

sure - but I am /guessing/ that you are expected to write the decimal approximations of the answers.

Answer 18

No decimal answers, just exact...
\[y=\frac{ 43 \pm \sqrt{85} }{ 18 }\]

Answer 19

do you want me to check that or are you confident in your work?

Answer 20

ya im pretty sure, thanks

Answer 21

yw - and well done! :)