Solve nine times two 𝑥 plus three equals 51. Give your answer in its simplest form.
When we see the word “solve” here, what it’s telling us is that we want to find out what 𝑥 is equal to. What value of 𝑥 makes this statement true? And to do that, we need to isolate 𝑥 and get 𝑥 by itself. We have these brackets here. And in order to get rid of these brackets, we’ll need to expand. And that means we’ll multiply nine by two 𝑥 and by three.
Nine times two 𝑥 equals 18𝑥. Nine times three equals 27. And 18𝑥 plus 27 is equal to 51. If our goal is to isolate 𝑥, we need to subtract 27 from both sides. Plus 27 minus 27 cancels out. 51 minus 27 equals 24. And 18𝑥 is the same thing as saying 18 times 𝑥. And that means we can divide both sides by 18. 18𝑥 divided by 18 equals 𝑥, and 24 divided by 18 is 24 over 18.
Remember, we’re looking for an answer in simplest form. And to simplify 24 over 18, we notice that both of these values are divisible by six. 24 divided by six is four, and 18 divided by six is three. 𝑥 must be equal to four-thirds for this statement to be true.
But this is not the only way to solve this problem. Let’s consider a second method. This time, we won’t expand what’s in the brackets. Instead, we’ll note that nine times two 𝑥 plus three equals 51. And then we’ll divide both sides by nine. On the left, multiplied by nine and then divided by nine cancels out. And we’re left with two 𝑥 plus three. And on the right, we have 51 over nine.
In this case, both the numerator and the denominator are divisible by three. 51 divided by three is 17, and nine divided by three is three. And now we need to subtract three from both sides of the equation. Three minus three equals zero, and two 𝑥 equals 17 over three minus three.
To subtract three from 17 over three, we need to rewrite this whole number as a fraction with a denominator of three. The whole number three equals the fraction nine over three. And now that we have this new fraction with a denominator of three, we can subtract 17 minus nine, which equals eight. And the denominator stays the same. Two 𝑥 equals eight-thirds.
And to isolate 𝑥, we divide both sides by two. If we need to divide eight-thirds by two, that’s the same thing as multiplying eight-thirds by one-half, which will give us eight-sixths. Both the numerator and the denominator are even. So we divide them both by two. Eight divided by two is four, and six divided by two is three. And we confirm that 𝑥 is equal to four-thirds.