Problem 9 - Fundamentals of Mathematics

Question

Find the value of each expression. $$ -(5-x)^{2}+7(m-x)+x-2 m, \text { if } x=5 \text { and } m=5. $$

Step-by-Step Solution

Answer

The value of the expression is -5.

Step 1: Substitute the values into the expression

First, substitute the given values of \( x = 5 \) and \( m = 5 \) into the expression \(- (5 - x)^{2} + 7(m - x) + x - 2m\). This changes the expression to: \(- (5 - 5)^{2} + 7(5 - 5) + 5 - 2 imes 5\).

Step 2: Simplify the power term

Calculate \((5 - 5)^2\). Since \((5 - 5) = 0\), we have: \((5 - 5)^2 = 0^2 = 0\). So the expression becomes \(-0 + 7(5 - 5) + 5 - 2 imes 5\).

Step 3: Simplify the multiplication

Now calculate \(7(5 - 5)\), which is \(7 imes 0 = 0\). The expression now simplifies to \(-0 + 0 + 5 - 2 imes 5\).

Step 4: Perform remaining arithmetic operations

The expression reduces to \(5 - 2 imes 5\). Now multiply \(2 \times 5 = 10\). Thus, the expression becomes \(5 - 10\).

Step 5: Calculate the final result

Subtract \(10\) from \(5\) to get \(5 - 10 = -5\). The expression simplifies to \(-5\).

Key Concepts

Substitution in Algebra

Substitution in algebra involves replacing variables in an expression with their given numerical values. This is a crucial skill that allows us to evaluate expressions with specific values. In our exercise, we started with the expression \[-(5-x)^{2} + 7(m-x) + x - 2m,\]and were given specific values, where \(x = 5\) and \(m = 5\). By substituting these values into the expression, we transformed it into a purely numerical one:

• Replace \(x\) with 5, which gives us \(5 - 5\).
• Similarly, replace \(m\) with 5, giving us \(7(5 - 5)\) and \(-2 \times 5\).

Now, our task was simply to evaluate the numerical expression by following arithmetic operations, transitioning from an algebraic form to a numerical one, making further simplification possible. Keep in mind that correct substitution sets the stage for accurately solving the whole expression.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This often includes combining like terms, eliminating parentheses through distribution, and applying fundamental arithmetic operations. After substitution in our example, we reached:\[ -(0) + 7(0) + 5 - 10. \]This step required recognizing parts of the expression that reduce to zero:

• Anything multiplied by zero becomes zero, like \(7 imes (5 - 5)\) resulting in 0.
• \(-(5 - 5)^2\) also becomes zero because \(0^2 = 0\).

Thus, we eliminate these terms, simplifying our expression to \[ 5 - 10. \]Remember, simplifying makes the final arithmetic operations easier and prevents errors in calculations.

Arithmetic Operations

At the heart of expression evaluation are arithmetic operations: addition, subtraction, multiplication, and division. Once we've substituted and simplified the expression, we're left with the rather straightforward task of arithmetic. In this case, the expression reduced to\[ 5 - 2 \times 5. \]Here, we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction):

• First, calculate multiplication: \(2 \times 5 = 10\).
• Then perform subtraction: \(5 - 10 = -5\).

It's crucial to maintain the correct operations sequence to ensure accurate results. A mistake as simple as performing operations out of order can lead to a very different answer. Concluding with solid arithmetic techniques guarantees the accuracy of the entire problem-solving process.