SOLUTION: the line 2x+3y=1 intersects the curve x(x+y)=10 at A and B.Calculate the coordinates of A and of B.

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Question 9223: the line 2x+3y=1 intersects the curve x(x+y)=10 at A and B.Calculate the coordinates of A and of B.
Answer by ichudov(507) About Me  (Show Source):
You can put this solution on YOUR website!
From the first equation 2x+3y = 1, we get y+=+%281-2x%29%2F3. Substitute it into the second equation
x%2A%28x%2B%281-2x%29%2F3%29+=+10
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • This is an equation! Solutions: x=5,x=-6.
  • Graphical form: Equation x%2A%28x%2B%281-2x%29%2F3%29=10 was fully solved.
  • Text form: x*(x+(1-2x)/3)=10 simplifies to 0=0
  • Cartoon (animation) form: simplify_cartoon%28+x%2A%28x%2B%281-2x%29%2F3%29=10+%29
    For tutors: simplify_cartoon( x*(x+(1-2x)/3)=10 )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at x%2A%28x%2B%28highlight_red%28+1+%29-2%2Ax%29%2F3%29=10.
Moved 1 to the right of expression
It becomes x%2A%28x%2B%28-2%2Ax%2Bhighlight_green%28+1+%29%29%2F3%29=10.
Look at x%2A%28x%2B%28-2%2Ax%2B1%29%2F3%29=highlight_red%28+10+%29.
Moved these terms to the left highlight_green%28+-10+%29
It becomes x%2A%28x%2B%28-2%2Ax%2B1%29%2F3%29-highlight_green%28+10+%29=0.
Look at highlight_red%28+x%2A%28x%2B%28-2%2Ax%2B1%29%2F3%29+%29-10=0.
Expanded term x by using associative property on %28x%2B%28-2%2Ax%2B1%29%2F3%29
It becomes highlight_green%28+x%2Ax+%29%2Bhighlight_green%28+x%2A%28-2%2Ax%2B1%29%2F3+%29-10=0.
Look at highlight_red%28+x+%29%2Ahighlight_red%28+x+%29%2Bx%2A%28-2%2Ax%2B1%29%2F3-10=0.
Reduce similar several occurrences of highlight_red%28+x+%29 to highlight_green%28+x%5E2+%29
It becomes highlight_green%28+x%5E2+%29%2Bx%2A%28-2%2Ax%2B1%29%2F3-10=0.
Look at x%5E2%2Bhighlight_red%28+x%2A%28-2%2Ax%2B1%29%2F3+%29-10=0.
Expanded term x%2F3 by using associative property on %28-2%2Ax%2B1%29
It becomes x%5E2-highlight_green%28+x%2F3%2A2%2Ax+%29%2Bhighlight_green%28+x%2F3%2A1+%29-10=0.
Look at x%5E2-x%2F3%2A2%2Ax%2Bx%2F3%2Ahighlight_red%28+1+%29-10=0.
Remove extraneous '1' from product highlight_red%28+1+%29
It becomes x%5E2-x%2F3%2A2%2Ax%2Bx%2F3-10=0.
Look at x%5E2-highlight_red%28+x+%29%2F3%2A2%2Ahighlight_red%28+x+%29%2Bx%2F3-10=0.
Reduce similar several occurrences of highlight_red%28+x+%29 to highlight_green%28+x%5E2+%29
It becomes x%5E2-highlight_green%28+x%5E2+%29%2F3%2A2%2Bx%2F3-10=0.
Look at highlight_red%28+x%5E2+%29-highlight_red%28+x%5E2%2F3%2A2+%29%2Bx%2F3-10=0.
Eliminated similar terms highlight_red%28+x%5E2+%29,highlight_red%28+-x%5E2%2F3%2A2+%29 replacing them with highlight_green%28+%281-1%2F3%2A2%29%2Ax%5E2+%29
It becomes highlight_green%28+%281-1%2F3%2A2%29%2Ax%5E2+%29%2Bx%2F3-10=0.
Look at %281-highlight_red%28+1+%29%2F3%2Ahighlight_red%28+2+%29%29%2Ax%5E2%2Bx%2F3-10=0.
Multiplied numerator integers
It becomes %281-highlight_green%28+2+%29%2F3%29%2Ax%5E2%2Bx%2F3-10=0.
Look at %28highlight_red%28+1+%29-highlight_red%28+2%2F3+%29%29%2Ax%5E2%2Bx%2F3-10=0.
Added fractions or integers together
It becomes %28highlight_green%28+1%2F3+%29%29%2Ax%5E2%2Bx%2F3-10=0.
Look at highlight_red%28+%28highlight_red%28+1%2F3+%29%29%2Ax%5E2+%29%2Bx%2F3-10=0.
Remove unneeded parentheses around factor highlight_red%28+1+%29,highlight_red%28+1%2F3+%29
It becomes highlight_green%28+1+%29%2Fhighlight_green%28+3+%29%2Ax%5E2%2Bx%2F3-10=0.
Look at highlight_red%28+1+%29%2F3%2Ax%5E2%2Bx%2F3-10=0.
Remove extraneous '1' from product highlight_red%28+1+%29
It becomes x%5E2%2F3%2Bx%2F3-10=0.
Look at highlight_red%28+x%5E2%2F3%2Bx%2F3-10+%29=0.
Equation highlight_red%28+x%5E2%2F3%2Bx%2F3-10=0+%29 is a quadratic equation: 0.333333333333333*x^2+0.333333333333333*x-10 =0, and has solutions 5,-6
It becomes highlight_green%28+0+%29=0.
Result: 0=0
This is an equation! Solutions: x=5,x=-6.

Universal Simplifier and Solver


Done!


x=5,-6
Substitute it into the first equation y+=+%281-2x%29%2F3
x=5:
Solved by pluggable solver: SUBSTITUTION solver -- compute expression given values
Your Result:


YOUR ANSWER


  • Graphical form: %281-2x%29%2F3 simplifies to -3
  • Text form: (1-2x)/3 simplifies to -3
  • Cartoon (animation) form: simplify_cartoon%28+%281-2x%29%2F3+%29
    For tutors: simplify_cartoon( (1-2x)/3 )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at %281-2%2Ahighlight_red%28+x+%29%29%2F3.
Replaced x with 5
It becomes %281-2%2Ahighlight_green%28+5+%29%29%2F3.
Look at %28highlight_red%28+1+%29-highlight_red%28+2%2A5+%29%29%2F3.
Replace constants in expression: highlight_red%28+1+%29,highlight_red%28+-2%2A5+%29 with highlight_green%28+-9+%29
It becomes %28highlight_green%28+-9+%29%29%2F3.
Look at %28highlight_red%28+-9+%29%29%2F3.
Removed extra sign in front of -9
It becomes %28-highlight_green%28+9+%29%29%2F3.
Look at highlight_red%28+%28-9%29%2F3+%29.
Removed extra sign in front of -3
It becomes -highlight_green%28+3+%29.
Result: -3

Universal Simplifier and Solver


Done!


x=-6:
Solved by pluggable solver: SUBSTITUTION solver -- compute expression given values
Your Result:


YOUR ANSWER


  • Graphical form: %281-2x%29%2F3 simplifies to 13%2F3
  • Text form: (1-2x)/3 simplifies to 13/3
  • Cartoon (animation) form: simplify_cartoon%28+%281-2x%29%2F3+%29
    For tutors: simplify_cartoon( (1-2x)/3 )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at %281-2%2Ahighlight_red%28+x+%29%29%2F3.
Replaced x with -6
It becomes %281-2%2Ahighlight_green%28+-6+%29%29%2F3.
Look at %28highlight_red%28+1+%29-highlight_red%28+2%2A-6+%29%29%2F3.
Replace constants in expression: highlight_red%28+1+%29,highlight_red%28+-2%2A-6+%29 with highlight_green%28+13+%29
It becomes %28highlight_green%28+13+%29%29%2F3.
Look at highlight_red%28+%28highlight_red%28+13+%29%29%2F3+%29.
Remove unneeded parentheses around factor highlight_red%28+13+%29
It becomes highlight_green%28+13+%29%2F3.
Result: 13%2F3

Universal Simplifier and Solver


Done!


Answer: system%28+x=5%2C+y=-3%29 and system%28+x=-6%2C+y=13%2F3%29+