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Chapter 10: Problem 29

Solve the given equation. $$10-2 \sqrt{y}=4$$

Short Answer

Expert verified
y = 9

Step by step solution

01

Isolate the square root term

To isolate the square root term, subtract 10 from both sides of the equation: \[10 - 2 \sqrt{y} - 10 = 4 - 10\] This simplifies to: \[-2 \sqrt{y} = -6\]
02

Divide by the coefficient of the square root term

Divide both sides by -2 to isolate the square root term: \[\frac{-2 \sqrt{y}}{-2} = \frac{-6}{-2}\] This simplifies to: \[\sqrt{y} = 3\]
03

Square both sides to eliminate the square root

Square both sides of the equation to eliminate the square root: \[(\sqrt{y})^2 = 3^2\] This simplifies to: \[y = 9\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Terms
Understanding how to isolate terms is essential in solving equations, especially those involving square roots. In the given exercise, we needed to isolate the square root term to simplify the equation. To do this, we performed the following steps:
  • First, recognized the term involving the square root: \(-2 \sqrt{y}\).
  • Second, we moved all other terms to the opposite side of the equation. In this case, we subtracted 10 from both sides:
\[10 - 2\sqrt{y} - 10 = 4 - 10\]
This step helps in isolating the term with the square root, simplifying our further calculations.
The resulting equation after this step was: \(-2\sqrt{y} = -6\).
This method of isolating terms makes it easier to solve for the variable in any algebraic equation.
Equations
Equations form the basis of algebra, and solving them often requires manipulating them to isolate the variable. In the given problem, we had an equation involving a square root term. An effective approach is:
  • First, identify the equation: \[10 - 2\sqrt{y} = 4\].
  • After isolating the square root term, which we discussed earlier, the equation became: \(-2\sqrt{y} = -6\).
  • Next, to further simplify, we divided both sides by -2:
\[\frac{-2 \sqrt{y}}{-2} = \frac{-6}{-2}\].
This leaves us with: \(\sqrt{y} = 3\).Practicing these steps with different equations strengthens your problem-solving skills, ensuring you understand the foundational principles of tackling equations in algebra.
Algebraic Methods
Algebraic methods are techniques used to manipulate and simplify equations to find unknown values. In our exercise, after isolating terms and simplifying the equation, we applied an algebraic method to solve for the variable. The key method here involves dealing with square roots:
  • First, after obtaining the simplified equation: \(\sqrt{y} = 3\), we removed the square root by squaring both sides:
\[(\sqrt{y})^2 = 3^2\].
  • By squaring, we eliminate the square root, giving: \y = 9\.
This algebraic method鈥攕quaring both sides鈥攊s a powerful technique in solving equations containing square roots. It simplifies the equation, allowing you to find the value of the variable efficiently. Understanding and applying these methods enhances your ability to solve a wide range of algebraic equations.

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Most popular questions from this chapter

Reduce to lowest terms. $$\frac{12-\sqrt{20}}{10}$$

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Reduce to lowest terms. $$\frac{6+\sqrt{12}}{8}$$
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