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Chapter 5: Problem 3

In Exercises 3鈥�12, solve the equation. Check your solution. $$ \sqrt{5 x+1}=6 $$

Short Answer

Expert verified
The solution to the given equation is \(x = 7\).

Step by step solution

01

Isolate the Square Root

Since we want to solve for 'x', the first step is to isolate the square root. But, it's already isolated in our equation, so no change is needed here.
02

Get Rid of the Square Root by Squaring the Equation

To cancel the square root, both sides of the equation are squared, resulting in: \((\sqrt{5 x+1})^2 = 6^2\), which simplifies to \(5x + 1 = 36\).
03

Solving the Equation

Subtract 1 from both sides of the equation to get: \(5x = 35\). Finally, divide both sides by 5 to find 'x': \(x = 7\).
04

Checking the Solution

To verify our solution, substitute \(x = 7\) back into the original equation. This gives us \(\sqrt{5*7 + 1} = \sqrt{36} = 6\). Since both sides of the equation equal 6, our solution \(x = 7\) is indeed correct.

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Key Concepts

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

Square Roots
Square roots are the inverse operation of squaring a number. They are often encountered in equations needing solutions. When we see a square root in an equation, our goal is to simplify or remove it by squaring. This step is crucial when solving equations like \(\sqrt{5x+1}=6\). By squaring both sides, we effectively eliminate the square root to simplify the equation.
  • In our example, squaring both sides turns \(\sqrt{5x+1}=6\) into \(5x+1=36\).

  • Always handle square roots with care since squaring both sides can introduce extraneous solutions.

Understanding square roots, therefore, is about recognizing their role in equations and systematically removing them with confidence.
Algebraic Manipulation
Algebraic manipulation involves rearranging an equation to solve for a variable. Starting from a squared equation like \(5x+1=36\) requires simple algebraic steps to find the value of 'x'.
  • First, isolate the term with the variable by subtracting from both sides. Here, subtract 1 from both sides to get \(5x = 35\).

  • Then, solve for 'x' by dividing both sides by 5, resulting in \(x = 7\).

These steps are fundamental in algebra. They not only find solutions but also help in shaping a clear path through complex equations. Practice these techniques to become more proficient in handling any algebraic expressions.
Verification of Solutions
Verification is the final step to ensure your solution is correct. After solving the equation, plug your solution back into the original equation to check it fits.
  • In this case, substitute \(x=7\) back into the original equation: \(\sqrt{5*7 + 1} = 6\).

  • This simplifies to \(\sqrt{36} = 6\), verifying that the equation holds true.

Verification is a vital step. It ensures no errors were made and confirms the solution as correct. This habit builds confidence and reliability in problem-solving.

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Most popular questions from this chapter

\(f(x)=\sqrt{x-3}+3\)

\(f(x)=\sqrt[3]{x}, g(x)=-\sqrt[3]{x}-1\)

\(\frac{1}{2} x=y^2-4\)

\(f(x)=\sqrt[5]{x}, g(x)=\sqrt[5]{-32 x}+3\)

You and a friend are playing a number guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2 , and then add 3 . Your friend's final answer is 53 . What was the original number chosen? Justify your answer.
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