Problem 31 - Algebra 1: Concepts and Skills

Question

CHECKING SOLUTIONS OF EQUATIONS. Check to see if the given value of the variable is or is not a solution of the equation. $$ 2 y^{3}+3=5 ; y=1 $$

Step-by-Step Solution

Answer

Yes, y=1 is a solution to the equation \(2y^{3}+3=5\)

Step 1: Substitute the given value into the equation

Replace y with 1 in the equation \(2y^{3}+3=5\). This gives \(2(1)^{3}+3=5\) which simplifies to \(2*1+3=5\).

Step 2: Simplify the equation

Proceed to simplify the equation. This gives \(2+3=5\).

Step 3: Check if it is true

Check if the left side of the equation equals the right side. It gives true statement as \(5=5\).

Key Concepts

Substitution Method

The substitution method is a technique commonly used to solve and verify solutions to equations. It involves replacing a variable in an equation with a given specific value to check if it transforms the equation into a true statement. In the given exercise, we substitute the variable \( y \) with the value \( 1 \) into the equation \( 2y^{3} + 3 = 5 \).

• First, substitute \( y = 1 \) into the equation: \( 2(1)^{3} + 3 = 5 \).
• This helps us to determine if the given value keeps the equation balanced.

By using substitution, we essentially take the variable out of the equation and replace it with the proposed solution to see if the equation holds true. This is a vital step in verifying solutions because it allows us to check correctness directly.

Simplifying Equations

Simplifying equations is an essential technique used to make equations easier to understand and solve. After substituting the variable with the given value, our next step is a simplification. The goal here is to clean up the expression so we can clearly see if it leads to a true statement. This is done through basic arithmetic operations.

• In our example, after substitution, the equation becomes \( 2 \times 1 + 3 = 5 \).
• Simplify by performing the exponentiation and multiplication: \( 2 \times 1 = 2 \).
• Add the result to the remaining parts of the equation: \( 2 + 3 = 5 \).

The process of simplification breaks down complex expressions and helps in visualizing whether both sides of the equation are equal, which is key to verifying the correctness of a solution.

Verifying Solutions

Verifying solutions is the final, but most important step in determining if the given solution solves the equation. This step assesses the validity of our outcome after substitution and simplification.

• After simplification, we're left with the statement \( 5 = 5 \).
• Both sides of the equation are equal, confirming that our solution \( y = 1 \) is indeed correct.

Verification is crucial as it ensures the solution is not only mathematically sound but also applicable to the problem at hand. It gives us confidence that the operations performed earlier were accurate. Through consistent practice of verifying solutions, students enhance their problem-solving skills and ensure a strong mathematical foundation.