Problem 10 - Algebra 2

Question

Solve each equation. Check your solutions. \(34=|y-2|\)

Step-by-Step Solution

Answer

The solutions are \(y = 36\) and \(y = -32\).

Step 1: Understand Absolute Value

The absolute value of a number is its distance from 0 on the number line, regardless of direction. This means if \(|y - 2| = 34\), then there are two possibilities: \(y - 2 = 34\) or \(y - 2 = -34\).

Step 2: Solve the First Equation

Start by solving the equation \(y - 2 = 34\). To isolate \(y\), add 2 to both sides: \[y - 2 + 2 = 34 + 2\]Thus, \(y = 36\).

Step 3: Solve the Second Equation

Next, solve the equation \(y - 2 = -34\). Add 2 to both sides to solve for \(y\): \[y - 2 + 2 = -34 + 2\]Thus, \(y = -32\).

Step 4: Check the Solutions

Substitute \(y = 36\) back into the original equation: \[|36 - 2| = |34| = 34\]This checks out. Now check \(y = -32\): \[|-32 - 2| = |-34| = 34\]This also checks out, confirming both solutions are correct.

Key Concepts

Solving Equations

When tackling equations, you aim to find the value of the variable that satisfies the equation. Here, the equation is \( |y-2| = 34 \). The key point with absolute value equations is understanding that \(|x| = a\) implies two scenarios: either \(x = a\) or \(x = -a\).
This doubles the number of equations we need to consider.

• First, solve the positive equation: \(y - 2 = 34\). Add 2 to both sides to isolate \(y\): \(y = 36\).
• Next, solve the negative equation: \(y - 2 = -34\). Again, add 2 to both sides: \(y = -32\).

The solutions, \(y = 36\) and \(y = -32\), come directly from considering both possible interpretations of the absolute value equation. This method ensures we've comprehensively solved all potential solutions.

Checking Solutions

Confirming your solutions ensures they are correct and not a result of calculation errors. When you substitute back into the original equation, you're effectively verifying the work done in solving steps.
To check, substitute each value of \(y\) back into \(|y - 2| = 34\).

• For \(y = 36\), substituting gives \(|36 - 2| = |34| = 34\). This confirms \(y = 36\) is correct.
• For \(y = -32\), substituting gives \(|-32 - 2| = |-34| = 34\). This confirms \(y = -32\) is correct as well.

Checking solutions is a crucial step in the problem-solving cycle as it helps catch mistakes and affirms the solution's validity.

Absolute Value Properties

Absolute value can transform problems and solutions by focusing purely on magnitude, irrespective of sign. Here are the core properties that guide solving equations involving \(|x|\):

• \(|x| = a\) implies \(x = a\) or \(x = -a\). This stems from the absolute value definition as distance from zero.
• \(|x|\) is always non-negative since distance cannot be negative.
• Absolute value equations often yield two potential answers, allowing one to capture both positive and negative scenarios.

Mastering these properties aids in understanding how absolute value equations expand the scope of potential solutions and necessitate double-checking each possibility. Embracing these characteristics ensures all angle possibilities are explored, which is especially helpful in real-world contexts where conditions can vary.