Problem 51 - Elementary Algebra Within Reach

Question

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(\frac{-y}{x^{2}+y^{2}}\) (a) \(x=0, y=5\) (b) \(x=1, y=-3\)

Step-by-Step Solution

Answer

The solution for (a) is -0.2 and for (b) is 0.3.

Step 1: Evaluate expression for (a)

For the first set of values, \(x=0, y=5\), substitute these values into the expression: \(\frac{-y}{x^{2}+y^{2}} \Rightarrow \frac{-5}{0^{2}+5^{2}} \Rightarrow \frac{-5}{25} = -0.2\)

Step 2: Evaluate expression for (b)

For the second set of values, \(x=1, y=-3\), substitute these values into the expression: \(\frac{-y}{x^{2}+y^{2}} \Rightarrow \frac{-(-3)}{1^{2}+(-3)^{2}} \Rightarrow \frac{3}{10} = 0.3\)

Key Concepts

Variable Substitution

Variable substitution is a fundamental step in evaluating algebraic expressions. It involves replacing variables, usually represented by letters such as \(x\) or \(y\), with specific numerical values given in an exercise. This process helps us simplify expressions to numerical values, making it easier to analyze or conclude.
When substituting, key points to remember include:

• Ensuring that every instance of the variable in the expression is replaced correctly.
• Using parentheses around substituted values, especially negative numbers, to prevent errors in calculation.
• Carefully following the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Each variable in an expression has its role, and substituting them accurately is crucial in solving algebra problems.

Evaluating Expressions

After substituting the values for variables, the next step is to evaluate the expression. This means performing the arithmetic operations in the correct sequence to arrive at a final numerical result.
In the exercise provided, after variable substitution, we simplify using basic math operations.

• For example, once \(x = 0\) and \(y = 5\) are substituted into the expression \( \frac{-y}{x^2 + y^2} \), it becomes \( \frac{-5}{0^2 + 5^2} \) which simplifies to \(-0.2\).
• Similarly, for \(x = 1\) and \(y = -3\), the expression \(\frac{-(-3)}{1^2 + (-3)^2}\) leads to \(\frac{3}{10} = 0.3\).

Working methodically through these steps ensures accurate results, helping to reinforce your understanding of algebraic principles.

Algebra Problem Solving

Algebra problem solving involves a systematic approach to finding solutions to equations or expressions. It starts with understanding the problem and identifying the expression or equation that needs solving.
In these exercises:

• The first step is recognizing what the problem is asking for—evaluating the expression for specific values of \(x\) and \(y\).
• Next comes the substitution of values, as discussed earlier.
• Then evaluate the expression carefully, ensuring each step follows mathematically logical order and precision.

Algebra is often a building block for more advanced math. Mastering these fundamental skills—substitution and evaluation—is critical. Practicing these steps boosts confidence and enhances problem-solving abilities, crucial skills for tackling more complex algebraic equations in the future.