Problem 42 - Algebra 2 and Trigonometry

Question

In \(39-42,\) solve and check each equation. $$ y+\sqrt{20}=\sqrt{45}-2 y $$

Step-by-Step Solution

Answer

The solution is \( y = \frac{\sqrt{5}}{3} \).

Step 1: Isolate y

First, let's move all the terms involving \( y \) to one side of the equation. Start by adding \( 2y \) to both sides: \[ y + 2y + \sqrt{20} = \sqrt{45} \] Simplify the equation: \[ 3y + \sqrt{20} = \sqrt{45} \]

Step 2: Simplify Radical Terms

Simplify \( \sqrt{20} \) and \( \sqrt{45} \). \[ \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \] \[ \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \] Now substitute these back into the equation from Step 1. This gives: \[ 3y + 2\sqrt{5} = 3\sqrt{5} \]

Step 3: Solve for y

Subtract \( 2\sqrt{5} \) from both sides to isolate \( 3y \): \[ 3y = 3\sqrt{5} - 2\sqrt{5} \] Simplify the right side: \[ 3y = \sqrt{5} \] Now divide both sides by 3 to solve for \( y \): \[ y = \frac{\sqrt{5}}{3} \]

Step 4: Check Your Solution

Substitute \( y = \frac{\sqrt{5}}{3} \) back into the original equation to verify it satisfies the equation. Left side of the equation: \[ y + \sqrt{20} = \frac{\sqrt{5}}{3} + 2\sqrt{5} \] Right side of the equation: \[ \sqrt{45} - 2y = 3\sqrt{5} - 2\left(\frac{\sqrt{5}}{3}\right) = 3\sqrt{5} - \frac{2\sqrt{5}}{3} \] Simplifying both sides results in both equating to \( \frac{9\sqrt{5}}{3} \), thus verifying the solution.

Key Concepts

Solving Equations

When tackling an equation like the one given, our primary goal is to determine the value of the variable in question. The key steps involve rearranging the equation and performing operations that simplify the task of finding this variable.
Here's a simple plan for solving equations:

• Combine like terms if applicable, to consolidate the equation into its simplest form.
• Move all terms involving the variable to one side of the equation using addition or subtraction.
• If the variable is multiplied by a number or expression, divide both sides by that to solve for the variable.

Following these steps ensures a structured approach to solve any equation efficiently.

Simplifying Radicals

Radical expressions, or square roots, often appear in equations. Simplifying these radicals can help you work through the problem more easily. The concept involves rewriting the square root so that external multiplication is minimized.To simplify a radical expression:

• Identify perfect square factors within the radicand (the number under the square root).
• Express the radicand as a product of perfect squares and other factors.
• Simplify the perfect square factor into its base number, moving it outside the square root sign.

For example, simplifying \( \sqrt{20} \) involves factoring it into \( \sqrt{4 \times 5} \), which becomes \( 2\sqrt{5} \) after simplifying \( \sqrt{4} \) to \( 2 \).

Isolating Variables

To find the solution of an equation, it's crucial to isolate the variable. This process involves manipulating the equation such that the variable of interest is left alone on one side of the equation.Steps to isolate a variable:

• Add or subtract terms on both sides to bring terms involving the variable to one side.
• Perform any necessary mathematical operations (such as dividing or factoring out), ensuring that the variable stands alone.
• Remember that whatever operation you do to one side of the equation, you must do to the other to maintain equality.

Through isolation, you simplify the problem to a point where you can easily solve for the unknown, like finding \( y = \frac{\sqrt{5}}{3} \) in our original equation.

Radical Expressions

Working with radical expressions can seem challenging at first, but knowing how to handle and manipulate them will greatly simplify such tasks. A radical expression typically involves the square root of a number, and they frequently appear in equations. Key aspects of radical expressions:

• Simplifying radicals by factoring out perfect squares clears their complexity.
• Ensure proper handling of radicals during operations, treating them similarly to variables where like terms are grouped together.
• When checking solutions involving radicals, always back-substitute to confirm if the found value satisfies the original equation.
• Use consistent mathematical operations to ensure the radical terms are correctly managed during the equation-solving process.

Mastering these expressions involves practice and understanding factoring, making them an easily manageable part of algebra.