Algebra: Adding and subtracting fractions
Making fractions similar
We can make two fractions \[ similar by multiplying the denominators with each other: \[ Note that both fractions now have the same denominator \(\blue b \green d\) and are therefore similar. |
Example Make #\tfrac{\orange 2}{\blue x}# and #\tfrac{\purple 3}{\green y}# similar: |
Sometimes there are common factors, and we do not have to multiply both denominators by each other to find a new denominator. Then you are able to find a new denominator by multiplying by the missing factors in the denominator. |
Example Make #\tfrac{\orange{2}}{\blue{x y}}# and #\tfrac{\purple{3}}{\green{y z}}# similar: \[\dfrac{\orange{2}}{\blue{x y}}= \dfrac{\orange{2} \green{z}}{\blue{x y} \green{z}} \qqquad \dfrac{\purple{3}}{\green{y z}}= \dfrac{\purple{3} \blue x}{{\blue x \green{y z}}} \] |
After all, when making fractions similar we choose the new denominator as the multiplication of both denominators: #\left(z-7\right)\cdot \left(z+3\right)#.
For #\frac{6\cdot z}{z+3}# we find this new denominator by multiplying numerator and denominator by a factor #z-7#.
This gives: \[\frac{6\cdot z}{z+3}=\frac{6\cdot \left(z-7\right)\cdot z}{\left(z-7\right)\cdot \left(z+3\right)}\]
For #\frac{-2\cdot z}{z-7}# we find this new denominator by multiplying numerator and denominator by a factor #z+3#.
This gives: \[\frac{-2\cdot z}{z-7}=\frac{-2\cdot z\cdot \left(z+3\right)}{\left(z-7\right)\cdot \left(z+3\right)}\]
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