Problem 72 - Algebra 2

Question

Solve each equation. Check your solutions. \(|5 y-8|=12\)

Step-by-Step Solution

Answer

The solutions are \(y = 4\) and \(y = -\frac{4}{5}\).

Step 1: Understand the Absolute Value

The equation given is \(|5y - 8| = 12\). Absolute value represents the distance from zero, meaning it can be either positive or negative. Therefore, we need to set up two separate equations to consider both cases.

Step 2: Setup Two Equations

Since the absolute value can be positive or negative, we consider two scenarios: 1. \(5y - 8 = 12\)2. \(5y - 8 = -12\).

Step 3: Solve the First Equation

Start with the equation \(5y - 8 = 12\). Add 8 to both sides to isolate the term with \(y\): \[5y = 20\] Now, divide by 5 to solve for \(y\): \[y = 4\].

Step 4: Solve the Second Equation

Now, solve the second equation \(5y - 8 = -12\). Add 8 to both sides: \[5y = -4\] Divide by 5: \[y = -\frac{4}{5}\].

Step 5: Check Solutions

Substitute back the solutions into the original absolute value equation:- For \(y = 4\): \(|5(4) - 8| = |20 - 8| = |12| = 12\), which is correct.- For \(y = -\frac{4}{5}\): \(|5(-\frac{4}{5}) - 8| = |-4 - 8| = |-12| = 12\), which is also correct.

Key Concepts

Equation Solving

Solving equations, especially those involving absolute values, requires careful consideration of the nature of absolute values. Absolute value measures how far a number is from zero on a number line, not the direction. This unique property results in absolute value equations having two potential outcomes. To tackle such equations:

• Identify the absolute value in the equation. Remember, \(|x| = b\) means \(|x|\) can be \(b\) or \(-b\).
• Break the equation into two separate cases: one where the expression inside the absolute value equals the positive, and another where it equals the negative of the given number.
• Solve each equation as you would with any linear equation.

Approaching them with these steps ensures a complete solution and accounts for all possible cases.

Algebraic Expressions

Algebraic expressions form the backbone of many mathematical operations. When dealing with equations such as \(5y - 8\), understanding components like terms and coefficients is essential. Consider some fundamentals:

• Terms are the distinct parts separated by '+' or '-' signs. In \(5y - 8\), \(5y\) and \(-8\) are terms.
• Coefficients are numbers multiplied by the variables within terms. Here, \(5\) is the coefficient of \(y\).
• Manipulations involve operations on these terms to isolate variables, like adding or subtracting numbers from both sides of an equation, or multiplying or dividing terms to simplify the expression.

Mastering algebraic expressions means knowing these elements well and using them to rearrange and solve equations efficiently.

Checking Solutions

Once you obtain potential solutions for any equation, it is vital to verify them. This process ensures accuracy and confirms that solutions are valid within the initial equation context.For absolute value equations:

• Substitute each found solution back into the original equation.
• Calculate to see if the equation balances, which means the left side equals the right side.
• If both sides are equal for all solutions, they are correct.

Hence, when solving \(|5y - 8| = 12\), substitute \(y = 4\) and \(y = -\frac{4}{5}\): both result in true statements, confirming their correctness. In this way, double-checking solutions avoids errors and reassures their validity, leading to a sound understanding and perfect answers.