Problem 59 - Algebra A Combined Function

Question

\(9(3 x+1)=4 x-5 x\)

Step-by-Step Solution

Answer

The value of \(x\) is \(\frac{-9}{28}\).

Step 1: Simplify the Equation

First, distribute the 9 on the left side of the equation: \[ 9(3x + 1) \rightarrow 9 \cdot 3x + 9 \cdot 1 \]This results in:\[ 27x + 9 = 4x - 5x \]

Step 2: Combine Like Terms on the Right

Combine the terms on the right side of the equation:\[ 4x - 5x = -x \]This simplifies the equation to:\[ 27x + 9 = -x \]

Step 3: Add x to Both Sides

To eliminate the \(-x\) on the right side, add \(x\) to both sides:\[ 27x + x + 9 = 0 \]This results in:\[ 28x + 9 = 0 \]

Step 4: Subtract 9 from Both Sides

Subtract 9 from both sides to isolate the term with \(x\) on one side:\[ 28x + 9 - 9 = 0 - 9 \]This simplifies to:\[ 28x = -9 \]

Step 5: Solve for x

Finally, solve for \(x\) by dividing both sides by 28:\[ x = \frac{-9}{28} \]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that involves multiplying a single term across terms within parentheses. Understanding how this property works is essential for simplifying complex algebraic expressions.
In the original exercise, we encounter the expression \( 9(3x + 1) \). Applying the distributive property, we multiply 9 by each term inside the parenthesis, leading to:

• \( 9 \times 3x = 27x \)
• \( 9 \times 1 = 9 \)

This results in the output \( 27x + 9 \). The distributive property allows us to transform the equation into a form that is simpler to work with.
By practicing this property, you can more easily handle problems that include both multiplication and addition or subtraction within parentheses.

Combining Like Terms

Combining like terms is another crucial process in simplifying equations, especially linear equations. It involves grouping terms that have the same variable and combining their coefficients.
In our step-by-step solution, after applying the distributive property, combine like terms on the right side of the equation:

• \( 4x - 5x \)

This simplifies to \( -x \). Here, only the coefficients of the terms differ, making it essential to subtract or add these coefficients while keeping the variable unchanged.
After combining, the equation becomes \( 27x + 9 = -x \). Mastering this technique will help streamline your solution path by reducing the complexity of the equations you're working with.

Isolating Variable

Isolating a variable typically involves getting the variable you're trying to solve for on one side of the equation by itself. This sets the stage for solving the equation completely.
In the example problem, after simplifying through the distributive property and combining like terms, the next step involves eliminating the \(-x\) from the right side of the equation. This requires moving terms with the variable to the same side by:

• Adding \(x\) to both sides gives us: \( 27x + x + 9 = 0 \)

This simplifies the equation to \( 28x + 9 = 0 \). Successfully isolating the variable in this step simplifies the equation further, making it easier to finish solving.
Isolating the variable is a strategic move in algebra, aiding in setting the equation up for its final simplification.

Solving for x

Solving for \( x \) is the process of finding the value of the variable that satisfies the equation. After isolating the variable term on one side, the focus is on simplifying the equation to reach a solution.
Following our steps, we have the equation \( 28x + 9 = 0 \). The aim now is to solve for \( x \). We start by eliminating the constant from the left side by:

• Subtracting 9 from both sides to get \( 28x = -9 \)

Next, solve for \( x \) by dividing by the coefficient of \( x \):

• \( x = \frac{-9}{28} \)

This calculates the precise value of \( x \) that makes the equation valid. The process of solving for a variable entails reversing operations and reducing the equation step-by-step until the variable is completely isolated.