Problem 35 - Introductory Algebra for College Students

Question

Solve: $$8(2-x)=-5 x$$

Step-by-Step Solution

Answer

\(x = 16/3\)

Step 1: Apply the Distributive Law

The first step is to apply the distributive law, which states that multiplication distributes over addition and subtraction. This means that 8 should be multiplied with both, 2 and -x. The result is the equation: \(16 - 8x = -5x\).

Step 2: Combine Like Terms

Combine terms with x on both sides of the equation, by adding 8x to both sides. This yields: \(16 = 3x\).

Step 3: Solve for x

To solve for x, divide both sides of the equation by 3: \(x = 16/3\).

Key Concepts

Understanding the Distributive Property

The distributive property is an essential tool in algebra, particularly when dealing with expressions involving parentheses. In simple terms, this property allows us to multiply a single term with each term inside the parentheses.

For example, in the equation given in the exercise, 8 is multiplied by both 2 and \(-x\). Therefore, the expression \(8(2-x)\) expands to \(8 \times 2\) and \(8 \times (-x)\), resulting in the new equation:

• \(16 - 8x\)

This careful distribution sets the foundation, converting a complex equation into something manageable, ensuring clarity and simplifying further steps.

Combining Like Terms

Once you've applied the distributive property, the next step is to make the equation simpler by combining like terms. 'Like terms' are terms that involve the same variables raised to the same power. In the context of the exercise, it’s about bringing together terms involving \(x\).

Initially, the equation is \(16 - 8x = -5x\). To simplify, you need to gather all \(x\) terms on one side of the equation. This means we add \(8x\) to both sides:

• \(16 = 3x\)

Now, the equation becomes much clearer, consisting only of constants on one side and \(x\) terms on the other. This makes solving the equation straightforward.

Solving for x

The final step in solving our linear equation is isolating the variable \(x\). In our simplified equation \(16 = 3x\), the goal is to solve for \(x\) by isolating it on one side of the equation.

This can be achieved by dividing both sides of the equation by 3:

• \(x = \frac{16}{3}\)

Now, \(x\) has been isolated, and we have found its value. Dividing ensures the equation remains balanced, which is a crucial principle in solving equations. This method allows us to find the precise value of \(x\) needed to satisfy the original equation. Understanding this final step is key to solving any basic linear equation you encounter.