What is the value of 𝑛 that satisfies the equation three times 𝑛 minus three plus
seven times two 𝑛 plus one equals 11𝑛?
First, we’ll copy down the equation. And then, we’ll need to distribute this multiplied by three and multiplied by seven
across the parentheses. We’ll start with our three. Three times 𝑛 equals three 𝑛. Three times negative three equals negative nine. And then, we have seven times two 𝑛 which equals 14𝑛. And seven times one will equal seven. And we’ll bring down the 11𝑛. At this stage, we want to combine like terms.
Three 𝑛 and 14𝑛 are like terms. And that means we can add them together. Positive three 𝑛 plus 14𝑛 equals 17𝑛. Negative nine and positive seven are also like terms. They’re numbers without a variable. And so, we can add them together. Negative nine plus seven equals negative two. 17𝑛 minus two equals 11𝑛. At this point, we want to get our terms with the variable 𝑛 onto the same side of
the equation. We can do this in one of two ways.
We can subtract 11𝑛 from both sides of the equation. 17𝑛 minus 11𝑛 gives us six 𝑛. And 11𝑛 minus 11𝑛 gives us zero. If we subtracted 11𝑛 from both sides, we would get six 𝑛 minus two equals zero. We can also solve by subtracting 17𝑛 from both sides. 17𝑛 minus two minus 17𝑛 equals negative two. And 11𝑛 minus 17𝑛 equals negative six 𝑛. We would have negative two equals negative six 𝑛 or we would have six 𝑛 minus two
On the left equation, we would divide both sides by negative six to get our 𝑛 by
itself. Negative six 𝑛 divided by negative six equals 𝑛. And negative two over negative six is the same thing as positive two over six, which
can be simplified to one-third. Or on the right side, to get 𝑛 by itself, we add two to both sides. And we have six 𝑛 equals two. To get 𝑛 by itself, we divide both sides by six. And then, we have 𝑛 equals two over six, which we simplified to one-third. 𝑛 equals one-third.
If we wanted to check, we can take one-third and plug it back in for 𝑛. One-third minus three is one-third minus nine-thirds. Three written as a fraction of three is nine-thirds. One-third minus nine-thirds is negative eight-thirds. And if we want to multiply that by three, we get negative twenty-four thirds. Two times one-third equals two-thirds plus one. We can write one as three over three. Two-thirds plus three-thirds equals five-thirds. And we multiply that by seven which gives thirty-five thirds. And 11 times one-third equals eleven-thirds. Since all three of our fractions have a denominator of three, we add negative 24 and
positive 35. And it becomes a fraction over three. Negative 24 plus 35 equals 11. And that means 𝑛 equals one-third does satisfy this equation.