Problem 126 - Prealgebra and Introductory Algebra

Question

Evaluate the expression for the given values of the variables. \(x-(-y),\) for \(x=-3\) and \(y=9\)

Step-by-Step Solution

Answer

Performing the addition, we find the result of the expression to be \(6\).

Step 1: Identify the Variables

In the given expression, the variables are \(x\) and \(y\). We are given, \(x = -3\) and \(y = 9\).

Step 2: Substitute the Variable Values

Substitute \(x = -3\) and \(y = 9\) into the expression. This give us \(-3 - (-9)\).

Step 3: Evaluate the expression

The expression is simplified by understanding that subtracting a negative number is the same as adding a positive number. So the expression \(-3 - (-9)\) simplifies to \(-3 + 9\).

Key Concepts

Substitution in Algebra

Substituting values into algebraic expressions is a fundamental skill in solving algebra problems. It involves replacing the variables in an expression with the numbers provided. To correctly substitute, carefully match each variable with its given number.

For instance, in the expression \(x - (-y)\), where \(x = -3\) and \(y = 9\), you insert the numerical values in place of the corresponding variables, getting \(-3 - (-9)\) as a result. It's essential to maintain the operations as they are in the original expression while doing the replacement to preserve the equation's integrity.

This process of substitution gives a numerical expression that is ready for further simplification and eliminates the ambiguity that sometimes comes with handling variables.

Negative Number Operations

Operations involving negative numbers can sometimes be counter-intuitive, but with a set of simple rules, they become much easier to manage. When you subtract a negative number, it is equivalent to adding its positive counterpart. This is because a negative sign in front of a negative number flips its sign.

So, for our example \(-3 - (-9)\), you're actually performing the operation \(-3 + 9\). It's imperative to remember that two negatives make a positive when next to each other in an equation. This rule is the key to avoiding simple mistakes that can occur when dealing with subtraction and addition of negative numbers.

Simplifying Expressions

Once you've completed the substitution and identified the negative number operations, it's time to simplify the expression. Simplifying means to perform all the arithmetic operations to reduce the expression to its simplest form, or a single number if possible.

In our problem, after substituting and changing the double negative into a positive, we're left with \(-3 + 9\). Simplifying this means you combine the numbers, which, as dictated by the standard rules of addition, results in \(6\). The process of simplifying is fundamental as it leads to the final, most concise form of the expression, providing the solution to the problem.

Remember the Order of Operations

• Perform the operations inside a parenthesis
• Execute multiplication or division from left to right
• Complete addition or subtraction from left to right

Despite our example not involving complex order of operations, it's a critical concept to keep in mind for simplifying more intricate expressions.