Problem 77 - Algebra A Combined Function

Question

Evaluate \(x+y\) for the given replacement values. See Examples 20 and 21. $$ x=-20 \text { and } y=-50 $$

Step-by-Step Solution

Answer

The value of \(x+y\) when \(x = -20\) and \(y = -50\) is \(-70\).

Step 1: Understanding the Problem

You're given the problem to evaluate the expression \(x+y\) with the provided values \(x = -20\) and \(y = -50\). This means that you need to substitute these given values into the expression and find the sum.

Step 2: Substitution of Values

Start by substituting the given values of \(x\) and \(y\) into the expression \(x + y\). This gives you \( (-20) + (-50) \).

Step 3: Evaluating the Expression

Now, you have the expression \((-20) + (-50)\). In this step, you simply need to add the two numbers together. Adding two negative numbers involves adding the absolute values and then affixing a negative sign to the result: \(20 + 50 = 70\). Hence, \(-20 + (-50) = -70\).

Step 4: Final Solution Confirmation

Lastly, verify the correctness of your solution. Ensure the addition of negative numbers was performed correctly: adding the absolute values gives \(70\), and the sum is negative since both original numbers were negative.

Key Concepts

Understanding Substitution in Algebraic Expressions

Substitution is a key skill in algebra that allows us to replace variables in an expression with specific values. This technique simplifies the process of evaluating expressions, leading us directly towards the solution.

• Imagine you have a treasure map. The variables are like the 'X marks the spot' clues on the map, and the numbers are the exact coordinates! Substitution is just the act of putting those coordinates into the map and finding out where you end up.
• In our exercise, we're given the expression \(x + y\) with \(x = -20\) and \(y = -50\). By substituting, we replace \(x\) with \(-20\) and \(y\) with \(-50\), transforming our expression into \((-20) + (-50)\).

Use substitution to simplify expressions before calculating. This step gives you clarity and allows you to focus on solving with the correct values.

Working with Negative Numbers

Negative numbers can be tricky. They have different rules compared to positive numbers, especially when it comes to operations like addition or subtraction.

• Think of negative numbers as debts. If you owe \\(20 and borrow another \\)50, your total debt is \$70.
• In our exercise, after the substitution step, we have the expression \((-20) + (-50)\). Notice how both numbers are negative.

When adding two negative numbers, you add their absolute values. Imagine if both were positive. You would simply add them. Here you do the same but recognize the result as negative:

• Find the absolute values: \(20\) and \(50\).
• Add them together: \(20 + 50 = 70\).
• Affix a negative sign: so \(-20 + (-50) = -70\).

Handling negative numbers requires paying extra attention to signs. Always perform operations on their absolute values first, then apply the sign of the original numbers.

Mastering Addition of Numbers

Addition is one of the first math operations we learn, and it remains essential throughout algebra. Let's see it applied to negative numbers, enhancing our understanding.

• The word 'addition' might confuse when dealing with negatives, especially since we end up subtracting absolute values mentally in some steps.
• The key is treating 'minus' as simply a direction or sign rather than a separate operation.

For example, adding two negative numbers:

• Switch to their positive counterparts to simplify mentally.
• Add the absolute values like regular numbers: \(20 + 50 = 70\).
• Apply the negative sign post-summation as they were both negative originally.

Through practice, addition becomes straightforward. Whether numbers are positive or negative, breaking the process into manageable steps is vital. These skills not only help solve our current problem of \(-20 + (-50) = -70\), but also bolster your problem-solving toolkit for future challenges.