Problem 28 - Algebra 2

Question

Evaluate each expression if \(a=\frac{2}{5}, b=-3, c=0.5,\) and \(d=6\). \(\frac{2 b-15 a}{3 c}\)

Step-by-Step Solution

Answer

-8

Step 1: Substitute the Given Values

To begin, substitute the given values into the expression: \(a = \frac{2}{5}, b = -3, c = 0.5\). The expression is \(\frac{2b - 15a}{3c}\). Replacing \(a, b,\) and \(c\) gives us:\[\frac{2(-3) - 15\left(\frac{2}{5}\right)}{3(0.5)}\]

Step 2: Simplify the Numerator

Next, focus on simplifying the numerator of the fraction: \[2(-3) - 15\left(\frac{2}{5}\right)\]Calculate each part step-by-step. First, \(2(-3) = -6\). Second, \(15 \cdot \frac{2}{5} = \frac{30}{5} = 6\). So, the numerator becomes:\[-6 - 6 = -12\]

Step 3: Simplify the Denominator

Now, simplify the denominator of the fraction:\[3(0.5) = 1.5\]

Step 4: Complete the Division

Now, combine the simplified numerator and denominator to complete the division:\[\frac{-12}{1.5}\]Perform the division: \[-12 \div 1.5 = -8\]

Step 5: State the Final Answer

The evaluated expression is \(-8\).

Key Concepts

Substitution

Substitution is a fundamental step in solving algebraic expressions. It involves replacing variables with their given numerical values. This makes the expression purely numerical, allowing for further operations. In this exercise, we were given specific values for the variables:

• - For \(a\): \(\frac{2}{5}\)
• - For \(b\): \(-3\)
• - For \(c\): \(0.5\)

By carefully substituting these values into the expression, \(\frac{2b - 15a}{3c}\), it transforms the expression into numerical terms, \(\frac{2(-3) - 15\left(\frac{2}{5}\right)}{3(0.5)}\).
This step is critical as it paves the way for performing arithmetic operations and ensures the problem is aligned with numerical rather than algebraic operations. To avoid mistakes, it's essential to carefully replace each variable with its respective value, maintaining parentheses to respect the order of operations where necessary.

Simplifying Fractions

Simplifying fractions is an essential skill in algebra. It involves reducing both the numerator and the denominator to their simplest form, ensuring the arithmetic operation can be completed easily and accurately. In the step-by-step solution, we first focused on the numerator: \[2(-3) - 15\left(\frac{2}{5}\right)\]
Let's break down this process:

• Calculate Each Term:
  • First, \(2(-3) = -6\)
  • Next, calculate \(15 \times \frac{2}{5}: \frac{30}{5} = 6\)
  By computing each part separately, you can see the expression is reduced to \(-6 - 6\) which simplifies to \(-12\).
• Simplifying the Denominator:
  • Multiply \(3 \times 0.5\) to get \(1.5\)

When both parts of the fraction have been independently simplified, completing the division \(\frac{-12}{1.5}\) becomes straightforward. Simplifying fractions efficiently is crucial to not only reach the correct answer but also to ensure each step is manageable.

Arithmetic Operations

Arithmetic operations form the core actions in evaluating an expression. They include addition, subtraction, multiplication, and division. After substitution and simplifying the fraction, completing the expression requires using arithmetic operations to finalize the calculation.
When tackling the expression \(\frac{-12}{1.5}\), you'll need to perform the division of these simplified terms. It can be helpful to visualize it as \(-12 \div 1.5\). The result of this division is \(-8\):

• Division Steps:
  • Think of \(-12\) being divided by \(1.5\) as finding how many times \(1.5\) fits into \(-12\). The negative sign indicates that the result will also be negative.

To check your work, remember that multiplying the result by \(1.5\) should return you to \(-12\). This cross-validation step confirms the arithmetic operations have been executed correctly. Mastering arithmetic operations ensures that evaluations and simplifications are correctly brought to a conclusion with precision and accuracy.