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How Do You Use a Solve a Polynomial Equation With Two Variables?

Solve for x and y in the following equation: 42xy+2x2y2=6x2y+14xy2

Summary

  1. Get all the terms on one side by subtracting 42xy and 2x2y2
  2. Factor out the GCF, 2xy
  3. Group together terms with common factors
  4. Factor out a common factor from each group, x from the first group and -7 from the second group
  5. Factor out a 3-y
  6. Split up the factors and set each one equal to 0
  7. Solve each equation for x and y

Notes

    1. We want to move all our terms to one side of the equation
    1. Since subtraction is the opposite of addition, subtracting 42xy and 2x2y2 from both sides will move them to the other side
    1. 6 factors into 2•3
    2. 14 factors into 2•7
    3. 42 factors into 2•3•7
    4. 2 is just 2
    5. These numbers have a common factor of 2
    1. So 2 will be a factor in our greatest common factor, or GCF
    1. Each term has at least one x, so x is a common factor
    1. So x will also be a factor in our GCF
    1. Each term has at least one y, so y is a common factor
    1. So y will also be a factor in our GCF
    1. Multiply our common factors of 2, x, and y to get the GCF
    2. The GCF is 2xy
    1. 6/2=3, and we can cancel an x and a y from 6x2y, leaving 3x
    2. 14/2=7, and we can cancel an x and a y from 14xy2, leaving 7y
    3. 42/2=21, and we can cancel an x and a y from 42xy, leaving 21
    4. 2/2=1, and we can cancel an x and a y from 2x2y2, leaving xy
    1. 3x and -xy have a common factor of x
    2. 7y and -21 have a common factor of 7
    1. If we group these terms together, we can factor out their common factors
    1. 3x-xy has a common factor of x
    2. 7y-21 has a common factor of 7
    1. The x's cancel in the first term, leaving 3
    2. The x's cancel in the second term, leaving y
    1. The -7's cancel in the first term, leaving y
    2. -21/-7 is 3
    1. Now we have another factor of 3-y that we can factor out
    1. We have x times 3-y and -7 times 3-y
    2. So 3-y is a common factor of these two terms
    1. Hooray!
    1. Now we can use the zero-product property to find our solutions
    1. We are multiplying three things together to equal 0
    2. So the zero product property says that one of those three things must equal 0
    3. We can use this fact to find our solutions
    1. Our factors are 2xy, 3-y, and x-7
    2. Set each of these factors equal to 0
    1. We can apply the zero-product property again here
    2. 2xy has 2 times x times y equal to 0
    3. So one of the factors must equal 0
    1. We already know that 2 doesn't equal 0, so we can ignore it
    2. But either x or y could equal 0 to give us an answer of 0
    3. So x=0 and y=0 are two possible solutions to our equation
    1. To solve our equation for y, undo the subtraction by adding y to both sides
    2. So y=3 is another possible solution that would make our factor 0
    1. To solve our equation for x, undo the subtraction by adding 7 to both sides
    2. So x=7 is another possible solution that would make our factor 0
    1. We found that x could either be 0 or 7
    1. Remember, braces are the curly brackets that contain sets
    1. Remember, braces are the curly brackets that contain sets
    2. We found that y could either be 0 or 3
    3. So these solutions go inside the braces
    1. We can go back and plug in each of the solutions we found to see if we get a true statement
    1. We need to look at x=0, x=7, y=0, and y=3
    1. When we put 0 in for x into our first factor, 2•0•y becomes 0
    2. Since now one of our factors is 0, our whole right hand side is 0
    3. Since we also have 0 on the left, we have a true statement!
    1. Multiplying by 0 makes everything 0
    2. So our whole right hand side equals 0
    1. So our solution of x=0 must be correct!
    1. When we put 7 in for x into our third factor, 7-7 becomes 0
    2. Since now one of our factors is 0, our whole right hand side is 0
    3. Since we also have 0 on the left, we have a true statement!
    1. Multiplying by 0 makes everything 0
    2. So our whole right hand side equals 0
    1. When we put 0 in for y into our first factor, 2•x•0 becomes 0
    2. Since now one of our factors is 0, our whole right hand side is 0
    3. Since we also have 0 on the left, we have a true statement!
    1. Multiplying by 0 makes everything 0
    2. So our whole right hand side equals 0
    1. When we put 3 in for y into our second factor, 3-3 becomes 0
    2. Since now one of our factors is 0, our whole right hand side is 0
    3. Since we also have 0 on the left, we have a true statement!
    1. Multiplying by 0 makes everything 0
    2. So our whole right hand side equals 0
    1. Since we got a true statement when we plugged in each of our values, we know our answers are correct!