Problem 66 - Prealgebra and Introductory Algebra

Question

Is 7 a solution of the equation \(-3 c=21 ?\)

Step-by-Step Solution

Answer

7 is not a solution to the equation \(-3 c=21\).

Step 1: Substitute the value

Substitute 7 for c in the equation. The equation then becomes \(-3(7)=21\).

Step 2: Apply operations

Apply the operation of multiplication. This results in \(-21=21\).

Step 3: Compare the results

Comparing both sides of the equation, \(-21\) does not equal 21.

Key Concepts

Substitution Method

The substitution method is a cornerstone of algebra that allows us to find the solution for an equation by replacing a variable with its actual value. When we have an equation like \(-3c=21\), we can evaluate whether a specific number is the solution by substituting it into the equation.

For instance, if we want to test if 7 is the solution, we replace 'c' with 7, resulting in \(-3(7)=21\). This process simplifies our job by turning a variable problem into a simple arithmetic check. Substituting correctly and evaluating the resultant expression are critical first steps to validating a proposed solution.

Multiplication in Equations

Multiplication is one of the basic arithmetic operations commonly used in solving equations. It is essential to understand how to multiply numbers and variables, as this skill is frequently involved when simplifying equations.

Using the equation \(-3c=21\) as an example, once 'c' has been substituted with 7, as shown in the substitution method, we then perform the multiplication: \(-3 \times 7\). Knowing that multiplying a negative number by a positive number results in a negative product is crucial. Therefore, the equation becomes \(-21=21\), which we can then analyze for correctness. Remember that abiding by the laws of arithmetic during multiplication is a key step in reaching the right solution.

Equation Solutions

An equation solution is the value of the variable that makes both sides of the equation equal. The ultimate goal of solving an equation is to find this exact value or determine if a proposed value is correct.

In the context of our example, after substituting and multiplying, we arrive at \(-21=21\). To determine if 7 is a solution to the equation \(-3c=21\), we must compare both sides of the equation. Here, we clearly see that \(-21\) is not equal to 21, thus 7 is not a solution to the equation. Understanding that solutions must equate and balance the equation reinforces the principle that for a value to qualify as a solution, it must satisfy the equation's original form.