Problem 9 - Algebra Form and Function

Question

In Exercises \(1-21,\) solve the equation for the variable. $$ \sqrt{y-2}=11 $$

Step-by-Step Solution

Answer

Answer: The value of y in the equation is \(y=123\).

Step 1: Square both sides of the equation

To eliminate the square root, square both sides of the equation: $$(\sqrt{y-2})^2 = 11^2$$

Step 2: Simplify the equation

Squaring both sides of the equation leads to: $$y-2 = 121$$

Step 3: Add 2 to both sides

Add 2 to both sides to isolate 'y': $$y = 121+2$$

Step 4: Calculate the value of 'y'

Compute the final value of 'y': $$y=123$$ The solution to the equation is \(y=123\).

Key Concepts

Understanding Square Roots

Square roots can often be a tricky concept, especially when you're new to solving equations. The square root is essentially the number that, when multiplied by itself, results in the original number. For instance, the square root of 25 is 5 because

• \(5 \times 5 = 25\).

When an equation involves a square root like \(\sqrt{y-2} = 11\), it's stating that 11, when squared, results in \(y - 2\). This has to be reversed if we want to find the value of \(y\).

To "unpack" the square root from the equation, we need to square both sides of the equation. This will result in:

• \((\sqrt{y-2})^2 = 11^2\), which further simplifies to \(y - 2 = 11 \times 11\).

This is a key step in solving any equation with a square root, as it helps to simplify the equation so you can figure out the variable's value.

Isolating the Variable

Once you have successfully eliminated the square root by squaring both sides of an equation, the next goal is to isolate the variable. Isolating the variable means getting the variable you are solving for, by itself, on one side of the equation. This helps determine its value directly.

For example, after addressing the square in \(\sqrt{y-2} = 11\), the equation simplified to \(y-2 = 121\). Here, isolating \(y\) involves adding 2 to both sides:

• \(y - 2 + 2 = 121 + 2\)
• This further simplifies to \(y = 123\).

This process lets us isolate \(y\) and find that it equals 123. It's the equivalent of "solving for \(y\)" because now \(y\) is clearly expressed in terms of other numbers with no additional operations required. This step is crucial in solving equations as it leads you directly to the answer.

Simplifying Equations

Simplifying an equation is an important technique that helps in making it easier to interpret and solve. Simplification techniques aim to reduce the equation to its most straightforward form while maintaining its equivalence. Simplifying can involve several methods based on what the equation requires, like combining like terms or simplifying expressions.

In the context of the problem \(\sqrt{y-2} = 11\):

• Initially, by squaring both sides, we simplified to \(y-2=121\).
• Then, by performing the operation of addition, we reached \(y=123\).

This two-part simplification starts by eliminating the square root, which is an obstacle to quickly assessing the relationship between the terms of the equation. Following this, numerical simplification further reduces the problem, leading directly to the solution. Simplifying ensures you're always working with the clearest possible expression, allowing for straightforward and accurate solutions.