Problem 42 - Prealgebra and Introductory Algebra

Question

Evaluate the variable expression for \(a=-2, b=4, c=-1,\) and \(d=3\) $$6 b \div(-a)$$

Step-by-Step Solution

Answer

The final result of evaluating the expression is \(24\).

Step 1: Substitution

Substitute the variables given into the expression. Substitute \(a=-2\) and \(b=4\) into the expression \(6b \div(-a)\) which gives the new expression \(6*4 \div(-(-2))\)

Step 2: Simplify the Negative

Address the double negative in the denominator of the expression. The rule of double negatives is two negatives make a positive, giving us the expression \(6*4 \div(2)\).

Step 3: Apply Division

Use division where specified in the expression. Now, the expression simplifies to \(6*4/2=12\)

Step 4: Apply Multiplication

Use multiplication to simplify the remaining parts of the expression. Finally, we get \(12*2=24\).

Key Concepts

Substitution in Algebra

In algebra, substitution is a fundamental skill that allows us to evaluate expressions by replacing variables with their actual values. It's like a mathematics version of fill-in-the-blanks, where the blanks are the variables. When you are given specific values for these variables, you can plug them into the expression to start making sense of it. It's important to substitute carefully to avoid errors. For example, if you are given the values for variables such as a=-2 and b=4, you will replace every instance of a with -2 and b with 4 in the expression. This transformation turns an abstract algebraic formula into a more concrete arithmetic problem, setting you up for the next steps: simplifying expressions and dealing with any negative numbers that might be present.

Simplifying Expressions

Once the variables in an algebraic expression are replaced with their numerical values, the next goal is to simplify the expression down to a single number if possible. This involves combining like terms, applying the order of operations (PEMDAS/BODMAS), and simplifying any arithmetic operations. Simplification is like cleaning up the clutter: it streamlines a bulky expression into something more manageable. In the context of the given exercise, once substitution is done, the next step is to resolve any double negatives and perform the necessary multiplication or division. It's essential to work methodically, ensuring that all operations are executed correctly and in the correct order. Simplifying can often reveal a more straightforward calculation hidden within a complex expression.

Negative Numbers in Algebra

Handling negative numbers can be tricky, but in algebra, there are clear rules to follow which makes dealing with them much easier. A critical point to remember is that a double negative results in a positive. Therefore, if you encounter a situation like (-a) where a is also a negative number, you would be dealing with a negative of a negative, which turns out positive. For instance, in the expression \(6b \/(-a)\), if \(a=-2\), the \( -(-a) \) becomes \( +2 \). Always pay careful attention to subtraction and division with negative signs because they're often the root cause of mistakes. Patience and practice with negative numbers build a strong foundation in algebra and empower students to handle more complex equations confidently.