Problem 102 - Algebra A Combined Function

Question

Name 2 numbers whose sum is -30

Step-by-Step Solution

Answer

The two numbers are -15 and -15.

Step 1: Define Variables

Let's start by defining two variables. We will denote the two numbers we are looking for as \( x \) and \( y \). Our task is to find two numbers such that their sum is equal to \(-30\). Thus, our equation is \( x + y = -30 \).

Step 2: Choose a Simple Number

To find two numbers, we can choose a simple number for \( x \). For example, let's assume \( x = -15 \).

Step 3: Solve for the Second Number

Substitute \( x = -15 \) into the equation \( x + y = -30 \) to find \( y \). This gives us:\[-15 + y = -30\]To solve for \( y \), add 15 to both sides of the equation:\[y = -30 + 15\]Thus, \( y = -15 \).

Step 4: Verify the Solution

Now, let's check to confirm that our solution is correct by substituting both values back into the original equation. We have \( x = -15 \) and \( y = -15 \):\[ x + y = -15 + (-15) = -30 \]This matches the original condition, so our solution is verified.

Key Concepts

Algebraic Variables

Algebraic variables are symbols used to represent unknown values in mathematical problems. In our exercise, we used the variables \( x \) and \( y \) to represent the two unknown numbers whose sum is \(-30\). By introducing variables:

• We can easily express complex problems in a simpler, more manageable form.
• Variables make it possible to solve equations where numbers are not immediately known.
• They help in setting up conditions, like an equation, based on known relationships.

Variables like \( x \) and \( y \) can take any numeric value. By providing a symbolic representation, variables help us organize information and logically solve problems through different mathematical operations.

Sum of Numbers

The concept of summing numbers involves adding them together to find a total. In this exercise, our goal was to find two numbers that sum up to \(-30\). Here’s why understanding sum is important:

• It is a fundamental operation in mathematics that can describe basic relationships between numbers.
• The sum gives a direct relationship that can be used to derive other unknowns in an equation.
• Understanding how sums work allows us to break problems into smaller, solvable parts.

For this example, given the equation \( x + y = -30 \), we needed to find combinations of \( x \) and \( y \) that satisfy this equality. Knowing one value allowed us to easily calculate the other and verify the solution.

Equation Solving

Equation solving is the process of finding the value of unknown variables that satisfy a given equation. It's a crucial skill in algebra and beyond. Let’s summarize the steps used in solving the equation in our problem:

• Define the Equation: Start by writing an equation that represents the problem. In our problem, it was \( x + y = -30 \).
• Choose a Value: Assign a value to one of the variables. We chose \( x = -15 \). This simplifies finding the second variable.
• Solve for the Other Variable: Substitute the known value into the equation and isolate the unknown variable. After substituting \( x \), we found \( y \) by calculating \( y = -30 + 15 \).
• Verify the Solution: Check that the values satisfy the original equation, ensuring everything adds up correctly. In this case, \( x + y = -15 + (-15)\) equaled \(-30\).

By following these steps, one can solve any linear equation with two variables, using logical reasoning and algebraic techniques.