Problem 14 - Thomas Calculus

Question

Solve the equations in Exercises \(13-18\) $$ |y-3|=7 $$

Step-by-Step Solution

Answer

The solutions are \(y = 10\) and \(y = -4\).

Step 1: Understanding the Problem

We are given the equation \(|y - 3| = 7\). This is an absolute value equation, where the absolute value of an expression equals 7. Our task is to find the value of \(y\) that satisfies this equation.

Step 2: Identifying Possible Cases

For an equation \(|A| = B\) to hold true, \(A\) must either be equal to \(B\) or \(-B\). Thus, we split our equation into two cases: 1. \(y - 3 = 7\)2. \(y - 3 = -7\)

Step 3: Solving Case 1: \(y - 3 = 7\)

To solve for \(y\), add 3 to both sides of the equation: \[y - 3 = 7\]\[y = 7 + 3\]\[y = 10\]

Step 4: Solving Case 2: \(y - 3 = -7\)

To solve for \(y\), add 3 to both sides of the equation: \[y - 3 = -7\]\[y = -7 + 3\]\[y = -4\]

Step 5: Conclusion

The solutions to the equation \(|y - 3| = 7\) are \(y = 10\) and \(y = -4\). Each value of \(y\) satisfies the original absolute value equation.

Key Concepts

Solving Equations

In mathematics, solving equations is a crucial skill, especially when dealing with different types of equations such as absolute value equations. An absolute value equation often appears when you're asked to solve |x - a| = b, where you need to determine the unknown variable that satisfies the equation. Let's break it down with our given problem, |y - 3| = 7. The absolute value of an expression is its distance from zero on the number line, irrespective of direction. To solve absolute value equations, you typically need to consider two separate cases because the expression inside the absolute value, here y - 3, can be either 7 or -7. This consideration helps in breaking down the problem into simpler linear equations that can be solved straightforwardly. Hence, regardless of whether the expression is positive or negative, its distance from zero is the same. This systemic breakdown allows you to find multiple potential solutions to the equation.

Mathematics Education

Mathematics education enriches a student's analytical abilities and problem-solving skills. One significant aspect of teaching mathematics is helping students understand and tackle absolute value equations. Through practice and exploration of problems like |y - 3|=7, students learn to approach complex scenarios with a structured method. This involves deciphering what absolute values represent and applying that knowledge to solve challenges.

• Training in strategizing both potential cases lays a foundation for critical thinking.
• It helps students parse expressions into manageable parts.
• Mathematics education emphasizes understanding concepts over memorizing formulas.

Consideration of different cases when dealing with absolute values teaches students valuable lessons about flexibility in thinking. As they encounter similar problems across other math domains, these foundational skills will serve them well.

Algebra Concepts

Algebra concepts form the backbone of solving absolute value equations. Understanding these concepts ensures a solid groundwork for approaching every algebraic expression and equation. Among them, the knowledge of evaluating expressions and equality principles can greatly aid in solving equations like |y - 3| = 7. Firstly, recognizing the format of |A| = B is crucial, indicating a need to assess both positive and negative possibilities for A. Fundamental principles of algebra, such as:

• Isolating the variable in linear equations,
• Balancing equations by performing congruent operations on each side,
• Breaking down complex expressions into simpler components,

all contribute to effectively solving these equations. Modern algebra education incorporates graphical, numerical, and analytical methods for deeper comprehension. By applying these algebraic techniques, students can engage more deeply with mathematical concepts, and their ability to solve various math problems is significantly enhanced.