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Chapter 10: Problem 14

Solve each equation. $$ \sqrt{5 x+1}-11=0 $$

Short Answer

Expert verified
The solution is \(x = 24\).

Step by step solution

01

Isolate the Square Root Term

Move all other terms to the right-hand side of the equation. \(\sqrt{5x + 1} = 11\)
02

Square Both Sides

Eliminate the square root by squaring both sides of the equation.\((\sqrt{5x + 1})^2 = 11^2\)This simplifies to\(5x + 1 = 121\)
03

Solve for x

Isolate x by subtracting 1 from both sides and then dividing by 5.\(5x = 120\)Divide both sides by 5.\(x = 24\)

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Key Concepts

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

Isolate the Terms
When solving a square root equation, your first task is to isolate the square root term. This means separating the part of the equation containing the square root from all other elements. For instance, in the equation \(\sqrt{5x+1} - 11 = 0\), you want to move the '-11' to the other side of the equation.
To achieve this, we'll add '11' to both sides:
\(\sqrt{5x+1} = 11\)
Now the square root term is isolated on one side of the equation.
Square Both Sides
Once you have isolated the square root term, the next step involves eliminating the square root symbol. You do this by squaring both sides of the equation. For example:
\[(\sqrt{5x + 1})^2 = 11^2\]
When you square a square root, you are left with the term inside the square root. Hence, the equation now becomes:
\[5x + 1 = 121\]
This step is crucial because it transforms a potentially complex problem into a simpler, linear equation.
Algebraic Manipulation
At this stage, you need to use basic algebraic operations to further simplify the equation. In our example, the equation is:
\(5x + 1 = 121\)
First, isolate the term containing the variable 'x' by subtracting '1' from both sides:
\(5x = 120\)
These simple operations continue to simplify the equation, bringing you one step closer to finding the value of 'x'.
Solve for x
The final step is to solve for the variable 'x'. Once you have simplified your equation to the form \(5x = 120\), the last operation is to isolate 'x'. You do this by dividing both sides by the coefficient of 'x', in this case, '5':
\(x = \frac{120}{5}\)
Perform the division to find:
\(x = 24\)
So, the solution to the equation \(\sqrt{5x + 1} - 11 = 0\) is \(x = 24\). Using these concepts step by step will simplify solving square root equations.

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