### 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. |
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. |
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(y+6\right)\cdot \left(y+7\right)#.

For #\frac{y}{y+6}# we find this new denominator by multiplying numerator and denominator by a factor #y+7#.

This gives: \[\frac{y}{y+6}=\frac{y\cdot \left(y+7\right)}{\left(y+6\right)\cdot \left(y+7\right)}\]

For #\frac{-9\cdot y}{y+7}# we find this new denominator by multiplying numerator and denominator by a factor #y+6#.

This gives: \[\frac{-9\cdot y}{y+7}=\frac{-9\cdot y\cdot \left(y+6\right)}{\left(y+6\right)\cdot \left(y+7\right)}\]

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