Problem 36 - Beginning and Intermediate Algebra

Question

Solve. $$\sqrt{5-x}=1$$

Step-by-Step Solution

Answer

The short answer to the given equation \(\sqrt{5-x}=1\) is to square both sides, simplify, and solve for \(x\), resulting in the solution \(x=4\).

Step 1: Square both sides of the equation

To remove the square root, square both sides of the equation: \((\sqrt{5-x})^2=(1)^2\)

Step 2: Simplify

Now, simplify both sides of the equation: \(5-x=1\)

Step 3: Add x to both sides

In order to isolate x, add x to both sides of the equation: \(5-x+x=1+x\) Which simplifies to: \(5=1+x\)

Step 4: Subtract 1 from both sides

Now, subtract 1 from both sides to completely isolate x: \(5-1=1+x-1\) Which simplifies to: \(4=x\) The solution to the given equation \(\sqrt{5-x}=1\) is \(x=4\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a crucial skill in solving equations, including those with square roots. It involves rearranging and simplifying equations to make them easier to solve. In this exercise, the main goal is to move between equivalent expressions to simplify the steps involved.
Benefits of algebraic manipulation:

• Allows us to convert complex expressions into simpler forms.
• Helps in isolating variables and finding solutions.
• Makes comparisons between different forms of equations easier.

In our solution, we used algebraic manipulation to move terms around the equation without changing the equality. For example, by adding \(x\) to both sides in step 3, we transformed the equation into a more manageable form. This process helps to focus only on the variable of interest and systematically simplifies the problem.

Isolation of Variable

Isolating the variable is a technique used to solve equations by getting the unknown variable on one side of the equation by itself. The goal is to have a simple expression where the variable equals a number.
To isolate the variable in our problem, we performed a series of operations:

• First, we added \(x\) to both sides to align the unknown on one side.
• Next, we subtracted 1 from both sides to isolate \(x\).

By consistently applying these operations, we simplified the equation and successfully isolated \(x\). Remember that whatever operation you do to one side of the equation, you must also do to the other. This keeps the equation balanced and true.

Squaring Both Sides

Squaring both sides of an equation is a common technique used to eliminate square roots. It works by raising both sides of the equation to the power of two, which in effect cancels the square root.
Steps involved:

• Identify the square root expression, \(\sqrt{5-x}\) in our case.
• Square both sides of the equation to remove the square root: \((\sqrt{5-x})^2 = (1)^2\).
• Simplify the squared terms: \(5-x = 1\).

This process allows us to transform the original equation into a polynomial equation that is easier to solve. However, it's crucial to check the solution afterward, as squaring can introduce extraneous solutions that don't satisfy the original equation.

Equation Simplification

Simplification is the key to making an equation more manageable. In solving square root equations, simplification usually follows squaring both sides. It involves reducing the equation to its simplest form.
In our exercise, after squaring, the simplification step involved:

• Directly reducing \(5-x = 1\) to the simplest forms of calculations possible.
• Step-by-step simplification by first isolating \(x\) and bringing the equation to \(5 = 1 + x\).
• Finally, subtraction to directly solve for the variable as \(x = 4\).

By breaking down the equation into simpler elements, each step becomes easier to handle, leading to a clear solution. Simplification ensures we can solve effectively and understand the logic behind each step.