Simplify the expression. 1 2 x 2 + 3 4 x y − 1 3 x 2 + 1 4 x y \frac12x^2+\frac34xy-\frac13x^2+\frac14xy

Here we're asked to simplify the expression one half x squared plus three fourths x y minus one third x squared plus one fourth x y. So starting with our steps here, we wanna go ahead and distribute any constants or variables into parentheses. In this particular expression, there's nothing that needs to get distributed. So we can move on to step two, identifying our like terms. Those terms that have the same variable raised to the same exponent. Looking at that very first term, one half x squared, so I have that variable x. It is raised to an exponent of two. And looking at the rest of my expression, I see another term, negative one third x squared. So that's the same variable x raised to the same exponent of two. So those are two like terms. Then I see this positive three fourths X Y. So there are two variables here, X and Y. Now looking at my last ex or my last term in my expression, positive one fourth X Y, this does still have the same variables, x and y, and they are raised to the same exponents. So x and y are both raised to an exponent of one. Remember that exponent is implied when there is no exponent. So those are another pair of like terms. So having identified our like terms, we can move on to grouping them. So one half X squared minus one third X squared, writing them next to each other. And then the same thing for my other two like terms, positive three fourths x y and then plus one fourth x y. So with our like terms grouped, we can now proceed with combining them by adding or subtracting their coefficients. Here we have coefficients that are fractions. So remember, we need to make sure that our fractions have like denominators before we can add or subtract them. So looking at these two terms here, one half x squared and negative one third x squared, these clearly have different denominators, two and three. Now in order to give them a like denominator, I actually need to change both of them. So I have one half and I have one third. In order to give these like denominators, I need to multiply one half by three over three and one third by two over two. So for one half times three over three, that's going to give me three sixths. And then for one third times two over two, this is going to give me two sixths, just multiplying across there. So these are my new coefficients with like denominators. So rewriting those up here, that one half became three sixths x squared and then minus one third that became minus two sixths. Now my next two like terms already have like denominators, so I'm going to go ahead and just rewrite them here to stay organized. Three fourths x y plus one fourth x y. So now everything that I need to have like denominators does and I can go ahead and just add or subtract accordingly. So three sixth minus two sixths is going to give me one sixth x squared and then three fourths plus one fourths gives me four over four x y. Now I can rewrite this still slightly more because four over four is just one. So I can rewrite this entire expression as being a one sixth x squared and then plus x y. That one is implied there. I don't have to worry about writing it. So this is my simplified expression, simplified from what we started with four terms to now two terms. Let us know if you have any questions and I'll see you soon.

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Simplify the expression.12x2+34xy−13x2+14xy\(\frac\)12x^2+\(\frac\)34xy-\(\frac\)13x^2+\(\frac\)14xy

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Simplify the expression.
$\frac{1}{2} x^{2} + \frac{3}{4} x y - \frac{1}{3} x^{2} + \frac{1}{4} x y$