Problem 24 - Algebra 1

Question

CHECKING SOLUTIONS OF EQUATIONS Check whether the given number is a solution of the equation. $$p^{2}-5=20 ; 6$$

Step-by-Step Solution

Answer

No, 6 is not a solution to the equation.

Step 1: Substitute the Given Number into the Equation

Replace \(p\) with 6 in the equation \(p^{2}-5=20\), so it becomes \((6)^{2}-5=20\)

Step 2: Simplify the Equation and Check the Equality

Calculate \((6)^{2}-5\) which equals 31. Then compare this value with the right side of the equation. Since 31 ≠ 20, 6 is not a solution to the equation.

Key Concepts

Substitution Method

The substitution method is a powerful technique in algebra used to determine whether a particular number is a solution to an equation. It involves substituting the given number into the equation in place of the variable, and then simplifying to see if the equation holds true.

In our exercise, we begin by substituting 6 for the variable \( p \) in the equation \( p^{2} - 5 = 20 \). This makes the equation \( (6)^2 - 5 = 20 \).

Let's break this down:

• First, substitute the variable with the provided number.
• Perform the necessary calculations to simplify the equation.
• Analyze whether both sides of the equation are equal after substitution.

The key goal of substitution is to either confirm or refute that the substituted value satisfies the original equation. In cases where the equality doesn't hold, it means the number is not a solution.

Checking Solutions

After substituting the number into the equation, the next logical step is to verify if the calculated values result in a true statement. This process is known as checking the solution. It's crucial because it confirms whether or not the chosen number is a valid solution to the equation.

In our exercise, once the substitution is made, you simplify \((6)^2 - 5\) which results in 31. Comparing this to the right side of the original equation, which is 20, you quickly see that these two values are not equal: 31 ≠ 20.

Here's how you check the solution:

• Calculate the expression on the left after substitution.
• Compare the result with the right side of the equation.
• If both sides match, the number is a solution. If not, it isn't.

Checking solutions ensures accuracy and helps avoid mistakes in the problem-solving process.

Algebraic Equations

Algebraic equations form the backbone of algebra. They are mathematical statements that show the equality between two expressions, usually involving variables and constants.

The equation in our example is \(p^{2} - 5 = 20\). This is a simple quadratic equation where \(p\) is the variable. Understanding the structure of algebraic equations is key to solving them effectively.

Key ideas about algebraic equations:

• Variables represent unknowns we aim to find or validate.
• Constants are known values, like the numbers 5 and 20 in our example.
• The goal is to determine the values of the variables that make the equation true.

In algebra, equations are solved by finding variable values that satisfy their equality; testing solutions is a crucial part of this process. Knowing the characteristics of algebraic equations helps in devising strategies like substitution and checking to find these solutions.