Problem 73 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is 0 a solution of \(x=5 x+15 ?\)

Step-by-Step Solution

Answer

No, 0 is not a solution to the equation.

Step 1: Substitute the Number into the Equation

Take the given number 0 and substitute it into the equation for x. This gives us the equation: \(0 = 5(0) + 15\).

Step 2: Simplify the Equation

Simplify the right side of the equation. Since \(5(0) = 0\), the equation simplifies to \(0 = 0 + 15\), which further simplifies to \(0 = 15\).

Step 3: Evaluate the Simplified Equation

Observe the simplified equation \(0 = 15\). Since 0 is not equal to 15, the equation is not satisfied when \(x = 0\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique used to solve equations by replacing variables with given values. It involves a few straightforward steps that make it quite approachable for beginners. In this method, we take a given number, usually specified in the problem, and replace the variable in the equation with this number.
This helps us determine if the given number is a solution to the equation.

• Start by identifying the variable in the equation.
• Replace the variable with the given number.

Once substitution is complete, we can move to the simplification step. Substitute, simplify, and validate - that's the essence of this method.

Simplifying Equations

Simplifying equations is an essential process in solving them effectively. After substitution, we focus on making the equation easier to work with by eliminating unnecessary complexities.
For example, in the equation:
\[0 = 5(0) + 15\]
First, simplify any calculations or operations. In this case:

• Calculate \(5 \times 0\), which equals 0.
• Substitute the result back into the equation to yield \(0 = 0 + 15\).
• Combine terms on the right-hand side to get \(0 = 15\).

This simplification process transforms the equation to a form that's easy to evaluate. It's all about making the equation clear and manageable.

Evaluating Equations

Evaluating equations is the final step where we decide if the equation holds true after simplification.
In our example, the equation simplified to:
\[0 = 15\]This step involves checking the balance of the equation. We have to determine if both sides of the equation are equal.

• Check if the left side equals the right side.
• If they are equal, the number is a solution; if not, it isn’t.

Since 0 is clearly not equal to 15, the number 0 is not a solution to the equation. Evaluating allows us to confirm the accuracy and truth of the equation under the given conditions.