Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, solve the equation for \(x\). $$ 3(2 x-1)+x=5 x+3 $$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Distribute and simplify

First, distribute the number 3 on the left-hand side of the equation. The equation becomes\[3(2x - 1) + x = 5x + 3 \]which simplifies to\[6x - 3 + x = 5x + 3\]Combine like terms on the left:\[7x - 3 = 5x + 3\]

Step 2: Move variables to one side

Next, to solve for \(x\), we want to get all \(x\) terms on one side. Subtract \(5x\) from both sides:\[7x - 5x - 3 = 3\]This simplifies to:\[2x - 3 = 3\]

Step 3: Isolate the variable term

Add 3 to both sides to isolate the term with \(x\):\[2x - 3 + 3 = 3 + 3\]This simplifies to:\[2x = 6\]

Step 4: Solve for x

Finally, divide both sides by 2 to solve for \(x\):\[\frac{2x}{2} = \frac{6}{2}\]This simplifies to:\[x = 3\]

Key Concepts

Distributive Property

When solving linear equations, the distributive property is often your first step. This property allows you to multiply a single term with each term inside a bracket. In the equation provided, 3 is distributed across the terms inside the parentheses: \(3(2x - 1)\). This becomes:

• \(3 \times 2x = 6x\)
• \(3 \times (-1) = -3\)

So, the expression \(3(2x - 1)\) simplifies to \(6x - 3\).
Using the distributive property correctly helps simplify the equation, making it easier to solve later steps. Remember to apply multiplication to all terms inside the parentheses.

Combining Like Terms

Combining like terms is about simplifying and consolidating the equation, particularly on one side. Like terms are terms that have the same variable raised to the same power. In the equation: \(6x - 3 + x = 5x + 3\), the terms \(6x\) and \(x\) can be combined, since they both involve \(x\).

• Add \(6x + x\) to get \(7x\).

This step reduces the expression to \(7x - 3\) on the left-hand side.
The simplified version of the equation is now easier to manage and allows us to move forward to isolating the variable.

Isolating Variables

Isolating variables means getting the variable by itself on one side of the equation. This step is crucial for finding its value. After combining the like terms, the equation is:\(7x - 3 = 5x + 3\). We want all terms containing \(x\) on one side. This is done by moving \(5x\) from the right to the left of the equation:

• Subtract \(5x\) from both sides to get \(7x - 5x\).

The equation simplifies to: \(2x - 3 = 3\). Adding 3 to both sides removes the constant term from the left:

• Add \(+3\) to both sides, simplifying to \(2x = 6\).

Now the variable \(x\) is isolated, meaning we can solve for its exact value.

Algebraic Manipulation

Algebraic manipulation is the final touch in solving for \(x\) in our equation. Once the variable \(x\) is isolated, meaning it's the only term on one side, we simply manipulate the equation to solve for \(x\). From the equation \(2x = 6\), we divide both sides by 2 to solve for \(x\):

• \(\frac{2x}{2} = \frac{6}{2}\)

This gives you \(x = 3\). Every algebraic step taken previously was leading up to this final manipulation to find the value of \(x\). Recognizing each operation and transformation in these steps is essential for accurately solving equations.