Problem 68 - Algebra 1: Concepts and Skills

Question

Check to see if the given value of the variable is or is not a solution of the equation. \(x+5=10 ; x=7\)

Step-by-Step Solution

Answer

No, the value \( x = 7 \) is not a solution in the equation \( x + 5 = 10 \).

Step 1 - Insert the variable value into the equation

Start by substituting the given value of the variable \( x \) into the equation. This means replacing \( x \) in the equation \( x + 5 = 10 \) with \( 7 \). Therefore, the equation becomes \( 7 + 5 = 10 \).

Step 2 - Perform the arithmetic calculation

Add up the values on the left side of the equation. When \( 7 \) is added to \( 5 \), the result is \( 12 \). So the equation after performing the operation will be \( 12 = 10 \).

Step 3 - Compare the results

Finally, compare the resulting values on both sides of the equation. In this case, \( 12 \) does not equal \( 10 \). Therefore, \( x = 7 \) is not a solution of the given equation \( x + 5 = 10 \).

Key Concepts

Variable Substitution

Variable substitution is a critical step when working with algebraic equations. It involves replacing the variable with a specific value provided or identified in order to simplify or solve the equation. In the context of checking whether a number is a solution to an equation, the process begins with taking the given value for the variable and plugging it into the equation where the variable appears.

For example, in the exercise provided, if we are given the equation \(x + 5 = 10\) and we want to check if \(x = 7\) is a solution, we substitute \(7\) for every instance of \(x\) in the equation. After substitution, our task is to verify if the equation holds true with the new numerical expression. This is a fundamental concept, as accurate substitution is the starting point for determining the validity of potential solutions.

Arithmetic Calculation

Arithmetic calculation is the process of performing basic mathematical operations such as addition, subtraction, multiplication, and division. After substituting the variable with its given value, it's essential to carry out the arithmetic calculation correctly to maintain the integrity of the equation.

In our example, following the variable substitution that leads to \(7 + 5\), we perform the addition to get \(12\). Good arithmetic skills ensure precise and efficient progression towards solving or checking an equation. It's one of the pillars of math problem-solving because, without accurate arithmetic, the subsequent steps and conclusions drawn would be flawed.

Equation Solving Steps

The process of solving an equation involves a sequence of steps that bring clarity and structure to the problem at hand. In general, solving steps can be outlined as follows:

• Identify the given equation and any values that are provided for the variables in the problem.
• Substitute the given value(s) into the equation as shown in the variable substitution concept.
• Execute the arithmetic calculations after substitution to simplify the equation.
• Analyze the resulting expression to determine whether it is true or false.
• Conclude whether the substituted values correspond to a solution of the equation.

In our given exercise, these steps are clearly applied and demonstrate that when applying arithmetic to the substituted values, we discover a mismatch – \(12\) does not equal \(10\). Thus, we can confidently state that \(x = 7\) is not a solution to the equation \(x + 5 = 10\). This step-by-step approach is critical for students to follow systematically in order to develop a solid understanding of equation solving.