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Consider doing the subtraction $11-25$. Often people do the easier subtraction $25-11$, get $14$, and then they put a negative in front of it to conclude $11-25=-14$. This works because

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\$11-25=-(25-11)=-14.\$

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So if we swap the order of subtraction, we need to put a negative out the front, but this is the same as just multiplying by $-1$ since $-(25-11)=-1\\times(25-11)$, which is also the same as dividing by $-1$.

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Therefore, reversing the order of a subtraction is the same as multiplying (or dividing) by $-1$.

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a)

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To determine if $\\frac{\\var{c}-x}{\\var{d}}$ is equal to $\\frac{x-\\var{c}}{\\var{d}}$, notice the only difference is the subtraction in the numerator is reversed. But $\\var{c}-x\\ne x-\\var{c}$. So these answers are not the same!

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To determine if $\\frac{\\var{c}-x}{\\var{d}}$ is equal to $-\\frac{x-\\var{c}}{\\var{d}}$, notice

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\\begin{align}-\\frac{x-\\var{c}}{\\var{d}}&=\\frac{-(x-\\var{c})}{\\var{d}}\\\\&=\\frac{-x+\\var{c}}{\\var{d}}\\\\&=\\frac{\\var{c}-x}{\\var{d}}\\end{align}

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So the negative out the front and the reversing of the subtraction cancelled each other out, and these answers are actually the same.

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b)

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You should notice that these fractions are very similar except that the order of subtraction is reversed in the numerator and the denominator. We should know that reversing the order of subtraction introduces a negative out the front, if we do this twice we will have two negatives out the front, which of course means a positive! That is,

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\\begin{align}\\frac{\\var{a}x^\\var{p}-\\var{b}y^\\var{q}}{\\var{c}xy-\\var{d}x^\\var{r}y^\\var{s}}&=\\frac{-(\\var{b}y^\\var{q}-\\var{a}x^\\var{p})}{\\var{c}xy-\\var{d}x^\\var{r}y^\\var{s}}\\\\&=\\frac{-(\\var{b}y^\\var{q}-\\var{a}x^\\var{p})}{-(\\var{d}x^\\var{r}y^\\var{s}-\\var{c}xy)}\\\\&=\\frac{\\var{b}y^\\var{q}-\\var{a}x^\\var{p}}{\\var{d}x^\\var{r}y^\\var{s}-\\var{c}xy}\\end{align}

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So the answers are the same!

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You should notice that in the numerator the order of subtraction has been swapped and in the denominator a $-\\var{d}x^\\var{r}y^\\var{s}$ has been replaced with $+\\var{d}x^\\var{r}y^\\var{s}$. These are not the same answers. If you require further proof, set them to be equal and see what happens, or even easier, substitute a value for $x$ and $y$ into both of them:

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Let $x=1$ and $y=1$ and we will compare the fractions. For 'your' answer we get $\\simplify[fractionNumbers,simplifyFractions]{{(a-b)/(c-d)}}$ but for 'your friends' answer we get $\\simplify[fractionNumbers,simplifyFractions]{{(a-b)/(c+d)}}$ and therefore the fractions are not equal!

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Students seem to freak out when their answer is not written exactly the same as the answer provided. This question tries to enforce that $(x-y)=-(y-x)$ and $\\frac{a-b}{c-d}=\\frac{b-a}{d-c}$

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equal!

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not equal!

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\$y=\\var{a*b}+\\frac{\\var{c}-x}{\\var{d}}.\$

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\$y=\\var{a*b} \\var{sym1}\\frac{x-\\var{c}}{\\var{d}}.\$

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equivalent! Just multiply (or divide) the numerator and denominator by $-1$ to see this.

", "

not equivalent!

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\$z=\\frac{\\var{a}x^\\var{p}-\\var{b}y^\\var{q}}{\\var{c}xy-\\var{d}x^\\var{r}y^\\var{s}}.\$

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\$z=\\frac{\\var{b}y^\\var{q}-\\var{a}x^\\var{p}}{\\var{d}x^\\var{r}y^\\var{s}\\var{sym2}\\var{c}xy}.\$

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These fractions are...

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This question asks you to compared different looking answers, and determine if they are equivalent.

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