document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #4964 by ichudov(507) $\"\"$ $\"About$

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\r
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Solved by pluggable solver: EXPLAIN simplification of an expression

Your Result:

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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.

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DETAILED EXPLANATION

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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 $\"x%2A%28x%2B%28-2%2Ax%2Bhighlight_green%28+1+%29%29%2F3%29=10\"$ .
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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 $\"x%2A%28x%2B%28-2%2Ax%2B1%29%2F3%29-highlight_green%28+10+%29=0\"$ .
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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 $\"highlight_green%28+x%2Ax+%29%2Bhighlight_green%28+x%2A%28-2%2Ax%2B1%29%2F3+%29-10=0\"$ .
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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 $\"highlight_green%28+x%5E2+%29%2Bx%2A%28-2%2Ax%2B1%29%2F3-10=0\"$ .
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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 $\"x%5E2-highlight_green%28+x%2F3%2A2%2Ax+%29%2Bhighlight_green%28+x%2F3%2A1+%29-10=0\"$ .
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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 $\"x%5E2-x%2F3%2A2%2Ax%2Bx%2F3-10=0\"$ .
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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 $\"x%5E2-highlight_green%28+x%5E2+%29%2F3%2A2%2Bx%2F3-10=0\"$ .
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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 $\"highlight_green%28+%281-1%2F3%2A2%29%2Ax%5E2+%29%2Bx%2F3-10=0\"$ .
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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 $\"%281-highlight_green%28+2+%29%2F3%29%2Ax%5E2%2Bx%2F3-10=0\"$ .
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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 $\"%28highlight_green%28+1%2F3+%29%29%2Ax%5E2%2Bx%2F3-10=0\"$ .
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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 $\"highlight_green%28+1+%29%2Fhighlight_green%28+3+%29%2Ax%5E2%2Bx%2F3-10=0\"$ .
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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 $\"x%5E2%2F3%2Bx%2F3-10=0\"$ .
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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: $\"0=0\"$
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Universal Simplifier and Solver

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Done!

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\n" ); document.write( "\n" ); document.write( "x=5,-6\r
\n" ); document.write( "\n" ); document.write( "Substitute it into the first equation $\"y+=+%281-2x%29%2F3\"$ \r
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Solved by pluggable solver: SUBSTITUTION solver -- compute expression given values

Your Result:

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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.

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DETAILED EXPLANATION

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Look at $\"%281-2%2Ahighlight_red%28+x+%29%29%2F3\"$ .
Replaced x with 5
It becomes $\"%281-2%2Ahighlight_green%28+5+%29%29%2F3\"$ .
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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 $\"%28highlight_green%28+-9+%29%29%2F3\"$ .
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Look at $\"%28highlight_red%28+-9+%29%29%2F3\"$ .
Removed extra sign in front of $\"-9\"$
It becomes $\"%28-highlight_green%28+9+%29%29%2F3\"$ .
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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: $\"-3\"$
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Universal Simplifier and Solver

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Done!

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\n" ); document.write( "\n" ); document.write( "x=-6:
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Solved by pluggable solver: SUBSTITUTION solver -- compute expression given values

Your Result:

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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.

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DETAILED EXPLANATION

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Look at $\"%281-2%2Ahighlight_red%28+x+%29%29%2F3\"$ .
Replaced x with -6
It becomes $\"%281-2%2Ahighlight_green%28+-6+%29%29%2F3\"$ .
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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 $\"%28highlight_green%28+13+%29%29%2F3\"$ .
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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: $\"13%2F3\"$
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Universal Simplifier and Solver

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Done!

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\n" ); document.write( "\n" ); document.write( "Answer: $\"system%28+x=5%2C+y=-3%29\"$ and $\"system%28+x=-6%2C+y=13%2F3%29+\"$ \n" ); document.write( "

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