How to solve abstract fractional (rational) equations
How do we solve rational equations?
At times we’d like to take an equation that has at least one fraction with a variable in its denominator and write the equation in a different way.
We’ll call an equation like this an abstract fractional equation. This lesson will look at how to do that.
There are a few things we want to remember about rational functions in general.
Multiplying a fraction by its reciprocal will always give you a value of ???1???.
For example ???x/y??? has a reciprocal of ???y/x??? because
???\frac{x}{y}\cdot\frac{y}{x}=1???
Remember that you can’t divide by ???0??? so some instructors might want you to include that ???x??? and ???y??? can’t equal ???0??? like this:
???\frac{x}{y}\cdot\frac{y}{x}=1??? where ???x,y\neq0???
2. To clear a fraction from an equation, multiply all of the terms on both sides of the equation by the fraction’s denominator.
For example, to clear the ???b??? from the fraction in
???ax+\frac{m}{b}=c???
multiply the equation by ???b??? on both sides.
???ax+\frac{m}{b}=c???
???b\left(ax+\frac{m}{b}=c\right)???
???abx+m=bc???
And in case your instructor wants to see it, remember you can’t divide by ???0???, so that means your new equation is true only where ???b\neq0???.
How to solve rational (abstract fractional) equations by multiplying by the least common multiple of all the denominators
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Solving the equation by clearing the denominators
Example
Solve the fractional expression for ???n???, if ???n\neq0???.
???\frac{m}{n}+x+ab=c???
In order to get rid of the fraction, we have to multiply every term on both sides of the equation by the denominator of ???m/n???.
???\frac{m}{n}+x+ab=c???
???n\left(\frac{m}{n}+x+ab=c\right)???
???n\cdot\frac{m}{n}+n(x)+n(ab)=n(c)???
???m+nx+nab=nc???
To solve for ???n???, we’ll need to collect all terms containing ???n??? on one side of the equation, and then factor out the ???n???.
Let’s move ???m??? to the right hand side and ???nc??? to the left.
???nx+nab-nc=-m???
Now factor out ???n???.
???n(x+ab-c)=-m???
Divide both sides by ???(x+ab-c)???.
???n=\frac{-m}{x+ab-c}???
Write the negative sign in front.
???n=-\frac{m}{x+ab-c}???
Let’s try another one.
Example
Solve for ???x??? if ???x\neq0??? and ???y\neq0???.
???\frac{1}{x}-\frac{m}{y}=p???
In order to get rid of the fractions, we have to multiply every term on both sides of the equation by both denominators, ???x??? and ???y???.
???\frac{1}{x}-\frac{m}{y}=p???
???xy\left(\frac{1}{x}-\frac{m}{y}=p\right)???
???xy\left(\frac{1}{x}\right)-xy\left(\frac{m}{y}\right)=xy(p)???
???1y-mx=xyp???
???y-mx=xyp???
To solve for ???x??? we’ll need to collect all terms containing ???x??? on one side of the equation, and then factor out the ???x???.
Let’s move ???mx??? to the right to get
???y=mx+xyp???
Now factor out the ???x???.
???y=x(m+yp)???
Divide both sides by ???m+yp???.
???\frac{y}{m+yp}=\frac{x(m+yp)}{m+yp}???
???\frac{y}{m+yp}=x???