In mathematics, letters from the Greek alphabet are often used as variables. Solve the following

In mathematics, letters from the Greek alphabet are often...

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of a variable that makes the equation true. Techniques such as distributing multiplication over addition, combining like terms, and isolating the variable on one side of the equation are key steps in the process. These methods are used across various branches of mathematics to systematically reduce and solve equations.

Greek Letters as Variables

In many areas of mathematics, variables are denoted by letters, and Greek letters are often used as alternatives to the standard Latin letters. This practice allows mathematicians to clearly differentiate between various types of quantities and can help in identifying constants, parameters, or specific variables in a given context.

Distributive Property

The distributive property is a fundamental algebraic rule that allows one to expand expressions where a multiplication operation is applied to a sum or difference inside parentheses. It states that multiplying a sum by a number yields the same result as multiplying each addend individually by the number and then summing the products.

Combining Like Terms

Combining like terms is an algebraic technique used to simplify expressions by summing coefficients of variables that have the same type and degree. This method is essential for reducing expressions to a simpler form, which makes solving equations more manageable by grouping similar terms together.

Transcript

00:01 In this problem, we're given the equation negative 7 times the quantity of alpha minus beta, minus 4 times the quantity of alpha minus theta, and it's equal to alpha divided by 2.

00:12 We're being asked to solve this for alpha.

00:14 Well, remember, alpha, beta, and theta are essentially, they're just variables.

00:18 They're like any of the other letters that we typically use.

00:21 So what we're going to do first is get rid of our parentheses by distributing.

00:25 So first we're going to distribute negative 7 to the first set of parentheses.

00:28 Well, negative 7 times alpha is just negative 7 alpha.

00:35 Then we need to multiply negative 7 times negative beta, which will become positive 7 beta.

00:42 Now we need to distribute the negative 4 to the second set of parentheses.

00:47 Well, negative 4 times alpha is negative 4 alpha, and negative 4 times negative theta is positive 4 theta.

00:59 And it's equal to alpha divided by 2.

01:02 So again, first thing, always get rid of your parentheses.

01:04 Now, the next thing i have is a fraction on the right -hand side of the equation.

01:09 I personally then like to get rid of fractions.

01:11 Well, on the right -hand side, we have alphas getting divided by 2.

01:15 Well, what's the opposite of dividing by 2? well, that would be to multiply by 2.

01:19 So what we're going to do is we're going to multiply both sides of our equation by 2.

01:25 So on the left -hand side, we're essentially going to distribute two to each of the terms.

01:30 Well, 2 times negative 7 alpha is negative 14 alpha.

01:37 2 times 7 beta is equal to 14 beta.

01:44 2 times negative 4 alpha is equal to negative 8 alpha, and 2 times 4 theta is equal to positive 8 theta.

01:59 Now, on the right -hand side of the equation, remember, the reason why we multiplied by 2, was to undo the division by two.

02:06 So these twos are essentially going to cancel each other out.

02:08 So we're just left with alpha...

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