Problem 73 - Beginning and Intermediate Algebra

Question

Solve. $$\sqrt{y+4}-\sqrt{y-1}=1$$

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{y+4}-\sqrt{y-1}=1\) is \(y = 5\).

Step 1: Isolate one of the square roots

We want to isolate one of the square roots, so let's start by isolating the \(\sqrt{y-1}\) term. To do this, simply add \(\sqrt{y-1}\) to both sides: \(\sqrt{y+4} = 1 + \sqrt{y-1}\)

Step 2: Square both sides of the equation

Now we can square both sides of the equation to eliminate the square root. \((\sqrt{y+4})^2 = (1 + \sqrt{y-1})^2\)

Step 3: Expand and simplify both sides of the equation

Expanding and simplifying, we get: \(y+4 = 1 + 2\sqrt{y-1} + (y-1)\) Now simplify further: \(y + 4 = y + 2\sqrt{y-1}\)

Step 4: Isolate the remaining square root

Subtract y from both sides of the equation: \(4 = 2\sqrt{y-1}\) Now divide both sides by 2: \(2 = \sqrt{y-1}\)

Step 5: Square both sides of the equation again

Once again, square both sides to eliminate the square root: \(2^2 = (\sqrt{y-1})^2\) So, \(4 = y-1\).

Step 6: Solve for y

Finally, add 1 to both sides of the equation to get: \(y = 5\) So, the solution to the equation \(\sqrt{y+4}-\sqrt{y-1}=1\) is \(y = 5\).

Key Concepts

Isolation of Square Roots

In radical equations, the goal is often to isolate a square root on one side. This makes the equation easier to work with. In our example, we start by isolating the \( \sqrt{y-1} \). This means moving \( \sqrt{y-1} \) to the right side, which simplifies our task.

• Move one square root to the other side by performing operations such as adding or subtracting.
• Once the square root stands alone, it becomes manageable to eliminate through subsequent steps.

In this equation, by adding \( \sqrt{y-1} \) to both sides, we turn the equation to \( \sqrt{y+4} = 1 + \sqrt{y-1} \). Keeping one square root isolated helps set up the next operations seamlessly.

Simplifying Equations

After isolating one of the square roots, we need to simplify the equations to facilitate further calculation. With our equation as \(([\sqrt{y+4}] = [1 + \sqrt{y-1}])\), simplifying becomes our focus.
This means expanding any binomials and reducing like terms.

• Use algebraic expansion techniques to manage expressions like \((1 + \sqrt{y-1})^2\).
• Simplifying involves combining like terms and preparing terms for further solutions.

In our example, when we expand, the terms \(y+4\) and \(2\sqrt{y-1}\) are the result. Recognizing patterns in terms is crucial for letting us proceed to the final solving steps.

Squaring Both Sides

Once a square root is isolated, another critical step in solving radical equations is squaring both sides. This operation removes the square root, transforming the equation into a more familiar format.

• Squaring every term on both sides ensures you maintain equation balance.
• Beware of potential extraneous solutions introduced during this step.

In our equation after the simplification to \(4 = 2\sqrt{y-1}\), squaring both sides turns it into \(4 = y-1\). As repetition of squaring steps can happen, double-check at each stage. This technique serves well in finally resolving \(y = 5\). Remember, the key is to maintain equation integrity without losing sight of the balance on both sides.