Problem 46 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(-2\left(\frac{11}{2}\right)(t)\) when \(t=-3\)

Step-by-Step Solution

Answer

The expression evaluated to 33 when \( t = -3 \)

Step 1: Substitute the value of \(t\)

First, replace \(t\) in the given expression, \(-2\left(\frac{11}{2}\right)(t)\), with its given value -3: \(-2\left(\frac{11}{2}\right)(-3)\)

Step 2: Simplify the expression

Next, perform the multiplication as follows:-2 * \( \frac{11}{2} * -3). The negative sign, '-2' cancels out with \(\frac{11}{2}\), which leaves us with 11. The multiplication of two negative numbers (-1*-3) gives you positive 3. So our expression now becomes 11 * 3

Step 3: Calculate the result

Now, calculate 11 times 3 which equals 33 : \(11 * 3 = 33 \)

Key Concepts

Multiplying Fractions

Multiplying fractions can look tricky at first, but it's quite simple once you understand the process. When you multiply fractions, you multiply the numerators together and the denominators together. For example, consider the expression:

• Multiply: \( \frac{a}{b} \times \frac{c}{d} \)

This means you multiply \(a\) with \(c\) and \(b\) with \(d\). The result is \( \frac{a \times c}{b \times d} \).

In a problem where you have a whole number multiplied by a fraction, like in the equation \(-2 \left( \frac{11}{2} \right) (-3)\), the approach is similar. The whole number is treated as a fraction by giving it a denominator of 1. Therefore,

• Convert whole number: \(-2 = \frac{-2}{1}\)

Now apply the rule of multiplying fractions and don't forget to handle the signs properly!

Substitution

Substitution is a key concept in algebra. It means replacing a variable in an expression with its given value and then simplifying the expression.

In our example, the expression \(-2 \left( \frac{11}{2} \right) (t)\) requires substituting \(t\) with its value of \(-3\). This changes the expression to:

• Replace \(t\) with \(-3\): \(-2 \left( \frac{11}{2} \right) (-3)\)

By substituting \(t\) with \(-3\), we are now ready to simplify and solve the expression. The key is to ensure the substitution is done accurately before proceeding with simplification. This step typically precedes any arithmetic operations.

Evaluating Expressions

Evaluating expressions is the process of finding the value of an expression for given variable values. After substitution, as seen in our earlier step, the next task is to simplify and perform the mathematical operations.

• Simplified expression: \(-2 \times \frac{11}{2} \times (-3)\)
• Calculate \(-2 \times \frac{11}{2}\), which simplifies to \(-11\), and \(-11 \times -3\) becomes \(11 \times 3\)

Perform the arithmetic to arrive at the final value. In this case, evaluating the expression yields \(33\).Understanding how to evaluate expressions by substituting values and performing operations in sequence is crucial in algebra. The use of order and careful arithmetic helps ensure accurate results.