Problem 67 - Beginning and Intermediate Algebra

Question

Decide whether the given number is a solution of the given equation. Is 0 a solution of \(2 x+6=5 x-1 ?\)

Step-by-Step Solution

Answer

0 is not a solution because 6 ≠ -1.

Step 1: Substitute the Number into the Equation

First, we substitute the given number, 0, for the variable x in the equation. Our equation becomes: \(2(0) + 6 = 5(0) - 1\).

Step 2: Simplify Both Sides of the Equation

Now, we simplify both sides of the equation. On the left side, 2 times 0 is 0, so we have 0 + 6 = 6. On the right side, 5 times 0 is 0, so we have 0 - 1 = -1. That leaves us with the simplified equation: \(6 = -1\).

Step 3: Compare the Two Sides

Lastly, we compare the simplified left side to the simplified right side. We see that 6 does not equal -1.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra, which involves finding the value of the variable that makes the equation true. When we talk about solving equations, we mean determining the values that satisfy the equation's balanced state. For example, in our original exercise, we have the equation \(2x + 6 = 5x - 1\). The aim is to figure out whether substituting 0 for \(x\) will make both sides of the equation equal.

Solving involves several methods and strategies like balancing operations, isolating the variable, and testing different values. The goal is to simplify the equation to the point where the solution is evident. It's akin to finding a key that fits a given lock to open a balanced state. By solving equations, you unlock information about relationships between quantities.

Substitution Method

The substitution method is a technique used to determine if a specific number is a solution to an equation. It involves replacing the variable in the equation with the given number and simplifying to see if the resulting statement is true. In our example, we substitute 0 into the equation \(2x + 6 = 5x - 1\), rewriting it as \(2(0) + 6 = 5(0) - 1\).

This method is straightforward as it allows us to test a specific value quickly. After substitution, we'll simplify the expressions to their basic form. If both sides of the equation equal, the substitution confirms the number as a solution. Otherwise, it is not a solution. This approach is particularly useful when checking if any provided value satisfies the given equation.

Comparing Expressions

Comparing expressions is the final step in checking solutions using substitution. After substituting the value and simplifying both sides of the equation, you compare the expressions to see if they are identical. In this context, we simplified the original equation to get \(6\) on the left and \(-1\) on the right.

If the expressions match, the equation holds true for the substituted value. If they do not, as in our example, where 6 does not equal -1, then the value is not a correct solution. This comparison completes the verification process. It ensures that the operations conducted, such as substitution and simplification, have been accurately performed, giving a clear and definitive answer to whether the initial number is a solution of the equation.