Problem 16 - Algebra 1

Question

CHECKING SOLUTIONS OF EQUATIONS Check whether the given number is a solution of the equation. $$4 c+2=2 c+8 ; 2$$

Step-by-Step Solution

Answer

No, 2 is not a solution to the equation \(4c+2=2c+8\).

Step 1: Substitute the number into the equation

Replace \(c\) in the given equation with the number 2. This gives: \(4(2) + 2 = 2(2) + 8\).

Step 2: Simplify the equation

Calculate the multiplication and addition operations on both sides of the equation. For the left-hand side: \(4(2) + 2 = 8+ 2 = 10\). For the right-hand side: \(2(2) + 8 = 4 + 8 = 12\).

Step 3: Compare both sides of the equation

After simplifying, we find that the left-hand side equals to 10 and the right-hand side equals to 12. Since the two sides don't equal each other, we can conclusively say that 2 is not a solution to the equation.

Key Concepts

Substitution

When checking if a given number is a solution to an equation, we begin with substitution. This involves replacing the variable in the equation with the specified number. For the equation provided in the exercise, the variable is represented by \( c \). The first step was to substitute \( c \) with 2. This transforms the equation into: \( 4(2) + 2 = 2(2) + 8 \).
Substitution allows us to see if the given number truly satisfies the equation. It's like testing the waters to see if the number fits.

• Locate the variable (in this case, \( c \)) within the equation.
• Replace all occurrences of the variable with the provided number (2).
• Maintain all other operators and constants as they are in the equation.

Understanding substitution is crucial because it sets the stage for verifying our number's validity as a solution.

Equation Simplification

After substitution, the next step is to simplify the equation. Simplification involves performing mathematical operations to unravel the expression into its simplest form. This includes completing multiplication and addition tasks. In our example:

• Left-hand side: \( 4(2) + 2 \) simplifies to 8 + 2, which equals 10.
• Right-hand side: \( 2(2) + 8 \) simplifies to 4 + 8, which equals 12.

Through simplification, we boil down the equation to its basic terms. This allows us to easily observe any differences or similarities between both sides of the equation.

Simplifying an equation correctly is important because any miscalculation can lead to incorrect conclusions about the validity of a solution.

Equality Comparison

With both sides of the equation simplified, the final step is to compare them. By this point, you'll have two numbers or expressions, one from each side of the equal sign. Here, the equation is simplified to determine that the left-hand side equals 10, while the right-hand side equals 12.

• If both sides match and are equal, the number is a solution to the equation.
• If they differ, as they do here, the number is not a solution.

Equality comparison is important because it checks the validity of our substitution and simplification steps. It's the moment of truth where we see the compatibility of the number with the original equation. In this example, since 10 is not equal to 12, we conclude that 2 is not a solution.