Question

Solve the equations exactly. Use an inverse function when appropriate. $$ \frac{3 \sqrt{x}+5}{4}=5 $$

Short answer

Question: Solve the equation for x: \(\frac{3\sqrt{x}+5}{4}=5\) Answer: \(x = 25\)

Step-by-step solution

Isolate the term containing the square root

Subtract 5 from both sides to isolate the term containing the square root: $$ \frac{3\sqrt{x} + 5}{4} - 5 = 0 $$ Now we have to find the common denominator which is 4, so rewrite 5 as \(\frac{20}{4}\): $$ \frac{3\sqrt{x} + 5}{4} - \frac{20}{4} = 0 $$ Subtract the fractions: $$ \frac{3\sqrt{x} + 5 - 20}{4} = 0 $$ Simplify the fraction: $$ \frac{3\sqrt{x} - 15}{4} = 0 $$

Clear the fraction

Multiply both sides of the equation by 4 to clear the fraction: $$ (4)\left(\frac{3\sqrt{x} - 15}{4}\right) = (4)(0) $$ Simplifying, we get: $$ 3\sqrt{x} - 15 = 0 $$

Isolate the square root

Add 15 to both sides of the equation to isolate the square root term: $$ 3\sqrt{x} = 15 $$ Now, divide both sides by 3: $$ \sqrt{x} = 5 $$

Eliminate the square root

Square both sides of the equation to eliminate the square root: $$ (\sqrt{x})^2 = 5^2 $$ Simplifying, we get: $$ x = 25 $$

Check the solution

Substitute x = 25 back into the original equation to check if the solution is correct: $$ \frac{3\sqrt{25}+5}{4}=5 $$ Simplify under the square root: $$ \frac{3(5)+5}{4}=5 $$ Multiply the numerator: $$ \frac{15+5}{4}=5 $$ Simplify: $$ \frac{20}{4}=5 $$ This simplifies to 5 = 5, so the solution is valid. Therefore, the solution to the equation is x = 25.

Key concepts

Inverse Functions and Their Role

In the world of algebra, an inverse function is essentially a function that "undoes" the work of the original function. When solving equations, using inverse functions can be a powerful tool. For example, if you are working with operations like addition, you can think of subtraction as the inverse, and vice versa.

• Multiplication and division are also inverse operations.
• Think of square roots and squares as inverses of each other.

In our exercise, after isolating the square root, we used the inverse function of squaring to clear it. Squaring both sides helped us find the value of \(x\), revealing that \(x = 25\). Understanding and applying inverse functions allow us to systematically solve equations, turning them into simpler forms that are easier to manage.

Isolating Variables to Solve Equations

Isolating variables is all about rearranging the equation to get the variable we are trying to find all by itself, usually on one side of the equation. This might involve a series of operations, achieved through logical steps.

• To begin with, isolate the term containing the variable.
• Use inverse operations to gradually eliminate other numbers or terms.

In this exercise, the equation initially involved fractions and a square root term. We started by isolating the square root \(3\sqrt{x}\) by removing fractions using the least common denominator. We then used further operations, such as addition and division, to completely isolate \(\sqrt{x}\), making it easier to proceed with the next steps.

Importance of Checking Solutions

Once a solution is found, it's crucial to check that it is correct by substituting it back into the original equation. This helps verify not only the accuracy but also ensures that there were no logical errors along the way.

• Always substitute the found value back into the original equation.
• Ensure that both sides of the equation balance.
• If both sides are equal, the solution is valid.

In this problem, after finding that \(x = 25\), we substituted it back into the original equation. By simplifying step-by-step, we confirmed that both sides equaled 5, verifying our solution. This reaffirmed that our steps and final answer were correct.

Simplifying Equations for Easier Solutions

Simplifying equations involves reducing them to their simplest form. This makes it easier to work with and solve them. Simplification is often a multistep process, involving several algebraic techniques.

• Eliminate fractions by multiplying both sides by the least common denominator (LCD).
• Combine like terms and perform operations to reduce complexity.

During this exercise, we eliminated fractions by multiplying through by 4, simplifying our task. We also combined like terms and used inverse operations to manage the equation better. By focusing on simplification, we make handling more complex algebraic equations less daunting, paving the way toward solving the problem effectively.