Problem 108 - Introductory Algebra for College Students

Question

Solve each equation in Exercises. $$2(3 x+4)=3 x+2[3(x-1)+2]$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\)

Step 1: Apply Distributive Law and Combine like terms

Applying the distributive property on both sides we get \(6x + 8 = 3x + 6x - 6 + 2\). Combine the similar terms to simplify the equation further: \(6x + 8 = 9x - 4\)

Step 2: Isolate the variable x

Isolate x by subtracting \(3x\) from both sides which gives, \(6x - 3x + 8 = 9x - 3x - 4\). Simplify to get: \(3x + 8 = 6x - 4\)

Step 3: Solve for x

Continue to isolate x by subtracting \(3x\) from both sides, this gives \(8 = 3x - 4\). Adding 4 to both sides gives the final solution: \(x = \frac{12}{3} = 4\).

Key Concepts

Understanding the Distributive Property

The distributive property is a key algebraic rule that allows us to simplify expressions and solve equations efficiently. It states that multiplying a sum by a number is the same as multiplying each addend of the sum by the number and then adding the products. For example, in the expression 2(3x + 4), we apply the distributive property by multiplying 2 with each term inside the parentheses, leading to 6x + 8. This property is crucial when dealing with algebraic equations that involve parentheses.

When you encounter an equation like 2(3x + 4) = 3x + 2[3(x - 1) + 2], your first step should be to distribute the numbers outside the parentheses across the terms inside. This breaks down complex parts and sets the stage for combining like terms, subsequently leading to the isolation of the variable.

Combining Like Terms

Once the distributive property has been applied, the next step in solving an algebraic equation is often to combine like terms. Like terms are terms that have the same variables raised to the same power. In our equation, after distributing we have 6x + 8 on one side and 3x + 6x - 6 + 2 on the other. Combining the like terms 3x and 6x gives us 9x, simplifying the equation to 6x + 8 = 9x - 4.

It's essential to identify and add or subtract coefficients of like terms to consolidate the equation, making it easier to solve. Remember, you can only combine terms with the same variable and exponent; coefficients and constants can be combined too, as they are like terms with an understood variable of x^0.

Isolating Variables to Solve Equations

To find the value of the variable, you must isolate it on one side of the equation. In the step 6x + 8 = 9x - 4, we isolate x by subtracting 3x from both sides to get 3x + 8 = 6x - 4. Continue isolating x by performing the same operation on both sides, which maintains the equation's balance. For instance, subtract 3x again to get 8 = 3x - 4, and then add 4 to both sides to fully isolate x, giving us x = 4 as the final solution.

The process of isolating the variable may involve adding, subtracting, multiplying, or dividing both sides of the equation by the same number. It's a methodical approach to arrive at the simplest form of the equation that provides the solution for the unknown variable. Always perform the inverse operation to what is attached to the variable, and do it to both sides to keep the equation balanced.