Problem

Solve $B=R-\frac{1}{16}(c-3 D)$ for $D$.

04:06

Problem 119 Hard Difficulty

In mathematics, letters from the Greek alphabet are often used as variables. Solve the following equation for $\alpha$ (read as "alpha"), the first letter of the Greek alphabet.
$$
-7(\alpha-\beta)-(4 \alpha-\theta)=\frac{\alpha}{2}
$$

Answer

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Precalculus

Algebra

Elementary and Intermediate Algebra

Chapter 2

Equations, Inequalities, and Problem Solving

Section 4

Formulas

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Video Transcript

in this problem. We're given the equation negative seven times the quantity of Alfa minus beta minus four times to quantity of Alfa minus data. And it's equal to Alfa, divided by two. We're being asked to solve this for Alfa. Well, remember, Alfa bathe her and data are essentially they're just variables. They're like any of the other letters that we typically use. So what we're gonna do first is get rid of our parentheses by distributing. So first, we're going to distribute negative seven to the first set of parentheses. Well, negative seven times. Alfa is just negative seven Alfa. Then we need to multiply negative seven times negative beta, which will become positive seven people. Now we need to distribute the negative four to the second set of prophecies. Well, negative four times Alfa is negative for Alfa and negative four times negative. Negative data is positive for data and it's equal to Alfa, divided by two. So again, first thing always get rid of your parentheses. Now, the next thing I have is a fracture on the right hand side of the equation. I personally then like to get with the fractions Well, on the right hand side we have Alpha's getting divided by two. Well, what's the opposite of dividing by two? Well, that would be two multiplied by two. So what we're gonna do is we're gonna multiply both sides of our equation by two. So on the left hand side were essentially going to distribute to each of the terms. Well, two times negative seven Alfa, isn't it getting 14 Alfa two times seven beta is equal to 14 beta. Two times negative for Alfa is equal to negative. Eight Alfa and two times for theta is equal to positive. Eight data now on the right hand side of the equation, Remember, The reason why be multiplied by two was to undo the division by two. So these twos are essentially gonna cancel each other out. So we're just left with Alfa. Hurry! Well, now, remember, our goal is to get Alfa by itself. Well, on the left hand side of the equation, we have the like terms of negative 14 Alfa and negative eight Alfa will negative 14 Alfa minus eight. Alfa is negative. 22 Alfa and then we'll bring down her plus 14 beta plus eight data is equal to health. Well, remember, our goal is to get Alfa by itself. So what I'm going to do next is move the negative 22 Alfa to the right hand side of the equation. So that way, the Alfa will already be all by itself on the right hand side. So to move. Negative 22 Alfa, I'm gonna add 22 Alfa to both sides of the equation. So when I do this first, no negative 22 Alfa plus 22 Alfa is zero. So those cancel each other out. So on the left hand side, unless, with 14 data plus eight data is equal to well, we have Alfa plus 22 Alfa will remember. There's an imaginary coefficient of a one in front of the Alfa. So one Alfa plus 22 Alfa would be 23 Alfa now, as you can see of ran out of room. So I'm going to start a new page. But I'm gonna keep this equation. So remember we had 14 beta plus eight data equal to 23 Alfa. Well, remember, we're trying to get Alfa by itself for right now. Alfa is getting multiplied by 23. So to undo multiplication we're going to divide. So we're gonna divide both sides of our equation by 23 now on the left hand side because we can't divide 14 beta or eight data both by 23 evenly. That's a simplified as it gets. So we're left with 14 beta plus eight data divided by 23 is equal to well, on the right hand side, we have 23 divided by 23 which is one, So they cancel l. And now we've got an Alfa all by itself. So what we found is that 14 beta plus eight favor divided by 23 is equal doubt.

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