Video Transcript
If 𝑚 plus three over 𝑚 minus three equals 10, what is the value of 𝑚?
To solve this problem, we need to get 𝑚 minus three out of the denominator. So we multiply the fraction 𝑚 plus three over 𝑚 minus three by 𝑚 minus three over one. However, if we do that to the left-hand side, we also need to do it to the right-hand side. On the left, 𝑚 minus three in the numerator times 𝑚 minus three in the denominator cancel out. And we’re left with 𝑚 plus three over one, which we’ll just write as 𝑚 plus three. On the right-hand side, we need to multiply 10 by 𝑚 minus three.
We’ll have to distribute this multiplication to the 𝑚 term and to the negative three. 10 times 𝑚 equals 10𝑚 and 10 times negative three equals negative 30. We now have a statement that says 𝑚 plus three equals 10𝑚 minus 30. We want to get the 𝑚s on the same side of the equation. So we subtract 𝑚 from both sides. And we’ll get three equals 10𝑚 minus one 𝑚 equals nine 𝑚 and bring down the minus 30. At this point, we’ll add 30 to both sides. Three plus 30 is 33. 33 equals 9𝑚. We can divide both sides by nine. And we get 33 over nine equals 𝑚 or the more common direction to write it 𝑚 equals 33 over nine. Both the numerator and the denominator are divisible by three. And we can simplify 33 over nine by dividing the numerator and the denominator by three, which will give us 11 thirds. 𝑚 equals 11 thirds.
Now, let’s check to see if this is true. We want to know if 11 thirds plus three over 11 thirds minus three equals 10. Let’s rewrite our threes as fractions with the denominator of three, which would be nine-thirds. In the numerator, we then have 11 thirds plus nine thirds and in the denominator, 11 thirds minus nine-thirds. Our numerator becomes 20 thirds and our denominator becomes two-thirds, which we’ll rewrite as 20 thirds divided by two-thirds. Instead of dividing by fractions, we’ll multiply by the reciprocal. Is 20 over three times three over two equal to 10? The threes cancel out and 20 over two does equal 10. And so, we can confirm that 𝑚 equals 11 thirds.
