Problem 58 - Elementary Algebra

Question

Solve using any method. $$ (3 x+5) 2-4=0 $$

Step-by-Step Solution

Answer

The solution is \( x = -1 \).

Step 1: Simplify the Equation

First, simplify the equation by distributing the 2 into the terms inside the parentheses. This gives us \( 2(3x + 5) - 4 = 0 \). Now multiply: \( 6x + 10 - 4 = 0 \).

Step 2: Combine Like Terms

Combine the constant terms. We have \( 10 - 4 \) which simplifies to 6. So the equation is now \( 6x + 6 = 0 \).

Step 3: Isolate the Variable

Move the constant term to the other side of the equation to isolate the variable. Subtract 6 from both sides: \( 6x = -6 \).

Step 4: Solve for x

Finally, solve for \( x \) by dividing both sides of the equation by 6. This gives \( x = \frac{-6}{6} \). Simplifying that, \( x = -1 \).

Key Concepts

Distributive Property

The distributive property is a powerful tool in algebra that allows you to expand or simplify expressions. It states that multiplying a single term by terms inside a parenthesis can be done separately. This means that if you have an expression like \[(a + b) \cdot c,\]you can distribute the \(c\) to both \(a\) and \(b\). This results in \[ac + bc.\]In our exercise, we needed to distribute a 2 across the terms inside \[(3x + 5) \cdot 2.\] By applying the distributive property, we have: * Multiply 2 by 3x to get 6x* Multiply 2 by 5 to get 10Thus, the expression becomes\[6x + 10.\] This step is essential before we can effectively continue with solving the equation. The main advantage here is that it removes the parentheses, simplifying the equation.

Isolation of Variables

Isolation of variables is a crucial step when solving linear equations. It involves manipulating the equation to get the variable, usually \(x\), on one side of the equation all by itself. Once alone, you can see the solution to the equation clearly. In our example, the equation is \[6x + 6 = 0.\] To isolate \(x\), you need to get rid of the constant next to it. Here's how:* Subtract 6 from both sides of the equation. This gives us\[6x = -6.\] Isolation makes it much easier to see what the next step is, leading directly to solving for the variable. Ensuring variables are isolated is a step you will encounter in almost every algebraic problem!

Simplification of Equations

Simplification of equations involves combining like terms and reducing expressions wherever possible. It makes equations easier to manage and solve and often happens concurrently with the steps above. In the problematic equation, after distributing, we formed:\[6x + 10 - 4 = 0.\] By combining like terms, specifically the constants, you simplify further:* Combine 10 and -4 to get 6.This simplifies the equation to:\[6x + 6 = 0.\] Ultimately, simplifying reduces errors in calculations and makes additional steps, such as isolation, more straightforward. It's a practice that helps in maintaining clarity and precision while solving complex problems.