Problem 90 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 10. $$ \begin{aligned} &(x-a)^{2}+(y-b)^{2} \text { for } x=-2, y=1, a=5, \text { and }\\\ &b=-3 \end{aligned} $$

Step-by-Step Solution

Answer

The expression evaluates to 65.

Step 1: Understand the Expression

The expression given is \((x-a)^2 + (y-b)^2 \)We need to substitute the values \(x = -2\), \(y = 1\), \(a = 5\), and \(b = -3\) into this expression.

Step 2: Substitute the Variables

Substitute \(x = -2\), \(y = 1\), \(a = 5\), and \(b = -3\) into the expression:\[((-2) - 5)^2 + (1 - (-3))^2\]

Step 3: Simplify Inside the Parentheses

Calculate the expressions inside the parentheses:\[((-2) - 5) = -7\quad \text{and} \quad (1 - (-3)) = 1 + 3 = 4\]

Step 4: Square the Simplified Terms

Square each of the results from Step 3:\[(-7)^2 = 49 \quad \text{and} \quad 4^2 = 16\]

Step 5: Add the Squared Values

Add the squared values from Step 4:\[49 + 16 = 65\]

Step 6: Final Step: Conclusion

The value of the expression \((x-a)^2 + (y-b)^2\) when \(x = -2\), \(y = 1\), \(a = 5\) and \(b = -3\) is 65.

Key Concepts

Substitution in Algebra

When working with algebraic expressions, substitution plays a crucial role. It's like replacing placeholders with actual values. In our case, the expression is \((x-a)^2 + (y-b)^2\). Each variable in this expression represents a particular number. By substituting, we are simply replacing these variables with given values.

For example:

• The variable \(x\) is replaced with \(-2\).
• The variable \(y\) with \(1\).
• The variable \(a\) with \(5\).
• The variable \(b\) with \(-3\).

Substitution helps to transform abstract expressions into numerical forms, making evaluation straightforward. With each value plugged in, the expression becomes specific, and we can easily proceed to manipulation.

Squaring Numbers

Once we've substituted the numbers into the expression, the next step is to square certain parts of it. Squaring a number is multiplying it by itself. This operation is common in expressions that involve exponents, particularly in quadratic equations.

Consider these examples:

• After substituting, we get \((-7)^2\). This means \(-7 \cdot -7\), which equals \(49\).
• The term \(4^2\) simplifies to \(4 \cdot 4\), equaling \(16\).

Squaring may change both the magnitude and the sign of numbers. Note: squaring any real number results in a non-negative outcome. Thus, knowing how to square properly ensures accurate expression evaluation.

Simplifying Expressions

Simplifying expressions is the final piece of the puzzle. Once substitution and squaring are done, you're left with basic arithmetic. Simplification means reducing an expression to its most concise form without changing its value. In our example, we deal with simple addition.

Here's how you simplify:

• You calculated \(49\) from \((-7)^2\).
• Next, you found \(16\) from \(4^2\).
• Finally, add these results: \(49 + 16\) resulting in \(65\).

Reduction of compound expressions or terms to simpler forms is vital in algebra. It not only makes solving exercises manageable but also reveals important relationships between components of the expressions.