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