Problem 29 - Fundamentals of Mathematics

Question

Find the value of each expression. $$5 a+6 b, \text { if } a=-6 \text { and } b=5$$

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Substitute the Given Values

Begin by substituting the given values of \(a\) and \(b\) into the expression. Here, \(a = -6\) and \(b = 5\). The expression is \(5a + 6b\). When you substitute the values, it becomes \(5(-6) + 6(5)\).

Step 2: Calculate Individual Terms

Simplify each term in the expression. Calculate \(5(-6)\) which results in \(-30\), and calculate \(6(5)\) which results in \(30\).

Step 3: Add the Results Together

Now add the results from the individual terms together: \(-30 + 30\).

Step 4: Simplify the Sum

Perform the addition \(-30 + 30\) to get the final result, which is \(0\).

Key Concepts

Substitution

Substitution in algebra is like swapping placeholders with their actual numbers. Imagine you have a math problem with letters, like trying to solve a puzzle. Each letter, or variable, stands for a number. In our exercise, we're told what these numbers should be. The variable \(a\) is given as \(-6\), and the variable \(b\) is \(5\). To find out what "5\(a\) + 6\(b\)" really means, we substitute. This means we replace the letter with its assigned number. When you substitute \(a\) with \(-6\) and \(b\) with \(5\), the expression changes to \(5(-6) + 6(5)\). By doing this, we transform a complex-looking expression into something we can easily calculate.

Simplifying Expressions

After substitution, the next step is simplifying the expression. Simplifying makes math problems shorter and easier to solve.First, handle the multiplication inside each term separately:- Calculate \(5(-6)\). Multiplying gives \(-30\) because a positive times a negative is negative.- Next, calculate \(6(5)\). The multiplication here is between two positives, so the result is \(30\).Now you have simplified each part of the expression to simple numbers: \(-30\) and \(30\). These are the new, clearer forms of your terms that show the calculated results.

Addition of Integers

Now, we need to add the simplified results. Both results are integers, which are whole numbers.Here's the simple yet important rule about adding integers:

• When numbers have different signs, subtract and keep the sign of the larger absolute value.

In our case, we have \(-30 + 30\). Since one number is negative and the other is positive, they cancel each other out. Think of it like levels on a number line. Going down 30 and then moving up 30 gets you right back to zero. Thus, when you add \(-30\) and \(30\), the sum is zero. This is because they perfectly balance each other out, resulting in a final answer of \(0\).