Problem 21 - Elementary Algebra

Question

Solve. $$ 4(3 x+5)=0 $$

Step-by-Step Solution

Answer

x = \frac{-5}{3}

Step 1: Distribute the Constant

Distribute the 4 to both terms inside the parenthesis: \[4(3x + 5) = 4 \times 3x + 4 \times 5\]This simplifies to: \[12x + 20 = 0\]

Step 2: Isolate the Variable Term

Subtract 20 from both sides of the equation to isolate the term with the variable:\[12x + 20 - 20 = 0 - 20\]Which simplifies to:\[12x = -20\]

Step 3: Solve for x

Divide both sides of the equation by 12 to solve for x:\[x = \frac{-20}{12}\]Simplify the fraction:\[x = \frac{-5}{3}\]

Key Concepts

distributive property

The distributive property is a fundamental concept in algebra. It allows you to multiply a single term by each term inside parentheses. This method helps simplify expressions and make them easier to work with. In our exercise, we applied the distributive property to the equation \(4(3x + 5) = 0\).

Here's how it works:

• Multiply the 4 by each term inside the parentheses: \(4 \times 3x + 4 \times 5\).
• This simplifies to: \12x + 20 = 0\.

Using the distributive property helps you expand and simplify equations in one clear step.

isolating the variable

Isolating the variable is an essential step in solving linear equations. It means getting the variable (in this case, \(x\)) alone on one side of the equation. This simplifies the equation and makes it easier to solve.

Let's see how we isolated the variable term in our example:

• We started with: \12x + 20 = 0\.
• To isolate \(12x\), subtract 20 from both sides: \12x + 20 - 20 = 0 - 20\, which simplifies to: \12x = -20\.

By isolating the variable, we made the next step—solving for \(x\)—much more straightforward.

simplifying fractions

Simplifying fractions is a key skill in algebra. It allows you to rewrite fractions in their simplest form, making them easier to understand and work with.

In our exercise, after isolating the variable \(12x = -20\), we needed to solve for \(x\). Here's how the simplification process works:

• Divide both sides by 12: \(x = \frac{-20}{12}\).
• Simplify the fraction: \(x = \frac{-5}{3}\).

Simplifying fractions ensures that your final answers are as clear and concise as possible.