Problem 17

Question

$$ x^{2 / 3}-y^{2 / 3}-2 y=2 ;(1,-1) $$

Step-by-Step Solution

Verified

Answer

The point (1, -1) satisfies the equation.

1Step 1: Substitute given point into the equation

We are given the equation $x^{2/3} - y^{2/3} - 2y = 2$ and the point $(1, -1)$. First, let's substitute $x = 1$ and $y = -1$ into the equation. This gives:\[ 1^{2/3} - (-1)^{2/3} - 2(-1) = 2 \].

2Step 2: Simplify the terms

Calculate each term individually: $1^{2/3} = 1$, $(-1)^{2/3} = 1$ (since $(-1)^2 = 1$ and $1^{1/3} = 1$), and the term $-2(-1) = 2$.

3Step 3: Resolve the equation

Plugging the simplified values back in, we get: \[ 1 - 1 + 2 = 2 \]. Simplifying the left-hand side gives \[ 2 = 2 \].

4Step 4: Verify the solution

Since the simplified left-hand side equals the right-hand side of the equation ($2 = 2$), the point $(1, -1)$ is a solution to the equation.

Key Concepts

Solving EquationsSubstitution MethodVerification of Solutions

Solving Equations

Solving equations in algebra involves finding the values for the variables that make an equation true. In our given exercise, we have the equation $x^{2/3} - y^{2/3} - 2y = 2$. This is a type of equation where we attempt to find values for $x$ and $y$ such that both sides of the equation are equal.

The key is to simplify the equation appropriately and test whether a given set of numbers, in this case, $(1, -1)$, satisfies the equation. Solving equations often involves:

Identifying the type of equation.
Simplifying to make the equation easier to work with.
Substitution of possible values to verify solutions.

Once the equation is simplified, you check your results to ensure accuracy, ensuring the solutions maintain the intended equality.

Substitution Method

The substitution method is a popular technique used in algebra to solve systems of equations or verify potential solutions. In the given problem $(1, -1)$ was substituted directly into the equation $x^{2/3} - y^{2/3} - 2y = 2$. By plugging $x = 1$ and $y = -1$ into the equation, we can see how these numbers interact within the equation.

Substitution effectively turns an algebraic equation into an arithmetic problem:

Find the expression or equation that needs solving, like $x^{2/3} - y^{2/3} - 2y = 2$.
Insert the known values into the equation: $x = 1$ and $y = -1$.
Simplify the equation by performing the necessary arithmetic operations.

In this exercise, this involved simplifying $1^{2/3} = 1$, $(-1)^{2/3} = 1$, and $-2(-1) = 2$, making it easier to evaluate whether the point satisfies the equation.

Verification of Solutions

Verification is a crucial step in algebra, ensuring the solution found is correct. In this exercise, after substituting $x = 1$ and $y = -1$ into the equation $x^{2/3} - y^{2/3} - 2y = 2$, we simplify and check if the left-hand side equals the right-hand side:

- Substituting the known values and simplifying, we got $1 - 1 + 2 = 2$.
- This result, $2 = 2$, confirms the point $1, -1$ is indeed a solution.

Verification includes checking all arithmetic steps to confirm that they contribute correctly to the solution:

Calculate each term with given values carefully.
Ensure the simplified expression maintains equality with the original equation's constraints.
Review calculations to avoid errors.

This step not only assures you of the solution's correctness but also helps reinforce understanding of the algebraic process.

Problem 17

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