Problem 49 - Algebra 1: Concepts and Skills

Question

Solve the equation. (Lesson 3.5) $$7 x-(4 x+3)=4(3 x+15)$$

Step-by-Step Solution

Answer

x = -60 / 9, so x = -20/3

Step 1: Distribute and Simplify

Distribute the values on both sides of the equation. This gives us: \(7 x - (4 x + 3) = 4(3 x + 15)\) which simplifies to \( 7x - 4x - 3 = 12x + 60\).

Step 2: Combine Like Terms

Combine the x terms on the left side of the equation and the constants on the right side of the equation. This yields: \(3x - 3 = 12x + 60\)

Step 3: Isolate x

Finally, we need to isolate x. First, subtract \(3x\) from both sides to get \(0 = 9x + 60\). Then, subtract 60 from both sides to get \(-60 = 9x\). Divide by 9 to get x on its own: \(-60 / 9 = x\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps to simplify expressions and solve equations. It states that multiplying a number by a sum is the same as multiplying each addend by the number and then adding the products. In the context of our equation, the distributive property is applied to both sides to eliminate parentheses from the expressions.
For example, consider the term on the left side:

• The expression \(7x - (4x + 3)\) becomes \(7x - 4x - 3\). This demonstrates distributing the \(-1\) across \(4x + 3\).
• On the right side, \4(3x + 15)\ becomes \12x + 60\. Here, you distribute the 4 across both \3x\ and 15, resulting in \4 \times 3x\ and \4 \times 15\.

Applying the distributive property correctly is crucial because it paves the way to simplify complex expressions, making it easier to solve the equation. Try practicing with different numbers to get the hang of it!

Combining Like Terms

Combining like terms is a strategy used to simplify expressions by adding or subtracting terms that have the same variable raised to the same power. This step makes expressions more manageable and helps in progressing toward the solution.
In our given problem, the equation separates terms with \(x\) from constant numbers:

• After applying the distributive property, the left side has the terms \(7x - 4x - 3\). Here, the like terms \(7x\) and \(-4x\) can be combined to result in \(3x\).
• This simplifies the equation to \(3x - 3 = 12x + 60\).

By grouping all terms containing the same variable together, you simplify the equation structure, preparing it for the variable isolation that follows. Always double-check to ensure you've combined terms accurately.

Isolation of Variables

Isolation of variables is the process of manipulating an equation in such a way that a particular variable stands alone on one side of the equation. This is a crucial step in solving linear equations as it provides the solution for the unknown variable.
In our equation, \(3x - 3 = 12x + 60\), the goal is to get \(x\) by itself:

• We start by subtracting \(3x\) from both sides to move all \(x\) terms to one side, resulting in \(0 = 9x + 60\).
• Next, subtracting 60 from both sides further isolates the variable term, giving us \(-60 = 9x\).
• Finally, divide each side by 9 to solve for \(x\), leading to the final expression \(x = -60 / 9\).

This step-by-step isolation method ensures the variable is neatly isolated, which makes identifying its value straightforward and easy to check back for accuracy in your original equation.