Problem 17 - Algebra 1

Question

CHECKING SOLUTIONS OF EQUATIONS Check whether the given number is a solution of the equation. $$2 y^{3}+3=5 ; 1$$

Step-by-Step Solution

Answer

Yes, the number 1 is a solution to the equation.

Step 1: Substituting y with the given number

Substitute y = 1 into the equation: \(2(1)^3 + 3\)

Step 2: Simplify the equation

Simplify the equation to get: \(2 + 3 = 5\)

Step 3: Checking the equality

As the equation holds true, it can be confirmed that y = 1 is indeed a solution for the given equation.

Key Concepts

Equation Verification

Whenever solving equations, it is crucial to verify if a proposed number is a solution to the equation. By doing this, we can confirm the accuracy of our solution. Equation verification involves evaluating a given expression by substituting the solution back into the original equation. This process helps to ensure that both sides of the equation are equal. If the equation holds true, the proposed number is indeed a solution.

To verify an equation:

• Substitute the potential solution into the original equation.
• Simplify the expressions on both sides of the equation.
• Compare the results from both sides. If they are equal, the number is a solution.

Let's say we have the equation \(2y^{3} + 3 = 5\) and the proposed solution is \(y = 1\). We substitute 1 for \(y\), and simplifying gives \(2(1)^3 + 3 = 5\). Since both sides equal 5, 1 is indeed a solution.

Substitution Method

The substitution method is a vital component in verifying solutions or solving systems of equations. This approach involves replacing a variable with a specific number or expression, which helps to simplify the equation and verify the correctness of a solution.

Here's how to apply substitution:

• Identify the variable you want to substitute in the equation.
• Replace the variable with its given value.
• Simplify the equation to evaluate if the expression holds true.

In our example, substituting \(y = 1\) into the equation \(2y^{3} + 3\) leads to the simplified form \(2(1)^{3} + 3\). Performing this step gives an immediate insight into the equation’s balance and whether the number is indeed the solution.

Cubic Equations

Cubic equations are polynomials of the third degree, with the general form \(ax^{3} + bx^{2} + cx + d = 0\). Solving cubic equations can sometimes be more complex than linear or quadratic equations due to their higher degree and additional variables. But don't worry, certain methods can simplify this process.

The given exercise presents a cubic equation \(2y^{3} + 3 = 5\), which is solved by isolating the cubic term.Characteristics of cubic equations:

• They have at least one real root.
• They can have either one or three real solutions based on their discriminant.
• The shape of their graph is a curve that may have inflection points.

To solve, cubic equations often involve factoring methods, synthetic division, or using the substitution method when specific values are provided, such as in our case where \(y = 1\). These steps simplify the equation to verify if the solution is applicable.