Problem 63 - Elementary Algebra

Question

Solve. $$ 12(3 x-1)+13(2 x-5)=0 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{77}{62} \).

Step 1: Expand the Equation

First, distribute the numbers outside the parentheses. This means multiplying each term inside the parentheses by the number outside.For the first term: \[ 12(3x - 1) = 12 \times 3x + 12 \times (-1) = 36x - 12 \]For the second term: \[ 13(2x - 5) = 13 \times 2x + 13 \times (-5) = 26x - 65 \]So, the equation becomes:\[ 36x - 12 + 26x - 65 = 0 \]

Step 2: Combine Like Terms

Combine the terms involving \(x\) and the constant terms. Combining \(x\)-terms:\[ 36x + 26x = 62x \]Combining constants:\[ -12 - 65 = -77 \]Now the equation is:\[ 62x - 77 = 0 \]

Step 3: Isolate the Variable

To solve for \(x\), we need to isolate it on one side of the equation. First, add 77 to both sides to get:\[ 62x = 77 \]Then, divide both sides by 62 to solve for \(x\):\[ x = \frac{77}{62} \]

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental tool in algebra that allows you to simplify equations. When you encounter an expression like \( a(b+c) \), the distributive property tells you that this is equivalent to \( ab + ac \). Essentially, you distribute the outer term to each term inside the parentheses. This property helps in expanding expressions and making them more manageable.In our exercise, the distributive property is applied to the terms \( 12(3x - 1) \) and \( 13(2x - 5) \). For the first term, we multiply 12 by both \( 3x \) and \( -1 \), yielding \( 36x - 12 \). Similarly, for the second term, multiply 13 by \( 2x \) and \( -5 \), resulting in \( 26x - 65 \). By distributing these numbers, we successfully break the expression down into simpler components.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. It involves adding or subtracting terms that share the same variable part. This helps streamline the expression into a simpler form.In our example, after using the distributive property, we ended up with terms like \( 36x \), \( 26x \), \( -12 \), and \( -65 \). To combine these, we first add \( 36x \) and \( 26x \) to get \( 62x \). Both of these terms involve the variable \( x \), making them like terms. Then, we combine the constant numbers \( -12 \) and \( -65 \) to get \( -77 \). By combining these terms, the equation transforms into a more workable form: \( 62x - 77 = 0 \).

Isolating Variables

Isolating variables is crucial when solving equations. This step involves arranging the equation so that the desired variable is alone on one side of the equation. The isolated variable then reveals its value based on the remaining terms.In our linear equation, \( 62x - 77 = 0 \), to isolate \( x \), we start by eliminating \( -77 \) from the left side. We do this by adding 77 to both sides, simplifying the equation to \( 62x = 77 \). Next, to get \( x \) by itself, we divide both sides by 62, resulting in \( x = \frac{77}{62} \). By isolating \( x \), we have successfully solved the equation, yielding the solution for the variable.