Question Video: Adding Algebraic Fractions with 𝑥 in the Numerator Mathematics

Video Transcript

Simplify two 𝑥 over five plus three 𝑥 over four.

In this question, we are asked to simplify the sum of two fractions. Since these fractions involve an unknown, 𝑥, we call these algebraic fractions. The techniques for adding algebraic fractions together are very similar to adding regular fractions. We need their denominators to be equal so that we can add them together by adding their numerators.

We can find that the lowest common multiple of the denominators is 20. So, we want to rewrite both fractions to have a denominator of 20. We can do this by multiplying the first fraction by four over four and the second fraction by five over five, where we can note that we are not changing the values in this expression since, in both cases, we are multiplying by one.

We now calculate that two 𝑥 times four is eight 𝑥 and three 𝑥 times five is 15𝑥. So, we are left with eight 𝑥 over 20 plus 15𝑥 over 20. Now that the denominators are equal, we can add the algebraic fractions by adding the numerators. We get eight 𝑥 plus 15𝑥 all over 20. We now want to add the two terms in the numerator. We see that these are like terms, so we add their coefficients to obtain 23𝑥 over 20.

Finally, since we need to write our answer in a simplified form, we need to check if there are any factors that we can cancel. We note that the highest common factor of 23 and 20 is one. So, we cannot simplify any further. Hence, we were able to simplify two 𝑥 over five plus three 𝑥 over four to be 23𝑥 over 20.