Problem 101 - Algebra A Combined Function

Question

Name 2 numbers whose sum is -17 .

Step-by-Step Solution

Answer

The two numbers are -10 and -7.

Step 1: Understanding the Problem

We need to find two numbers, say \( a \) and \( b \), such that their sum is -17. This can be written as the equation \( a + b = -17 \).

Step 2: Choose One Number

Let us assume one of the numbers, \( a \), and assign it a random value, for instance, \( a = -10 \). This is just a starting point to find a potential solution.

Step 3: Solve for the Second Number

Using the assumed value \( a = -10 \), we substitute into the equation \( a + b = -17 \) to find \( b \). Therefore, \( -10 + b = -17 \).

Step 4: Calculate the Second Number

Solve the equation \(-10 + b = -17\) by adding 10 to both sides: \( b = -17 + 10 \). This simplifies to \( b = -7 \).

Step 5: Verify the Solution

Verify that \( a = -10 \) and \( b = -7 \) indeed sum to -17: \(-10 + (-7) = -10 - 7 = -17\). Thus, the solution is correct.

Key Concepts

Linear Equations

When we talk about linear equations, we are referring to algebraic equations where each term is either a constant or the product of a constant and a single variable. These equations are essential in solving various basic math problems, as they enable us to express relationships between numbers in a simple, two-dimensional format. For example, in the expression \( a + b = -17 \), both \( a \) and \( b \) are variables that can be adjusted to find a solution.
Linear equations are called "linear" because they graph as a straight line when plotted on a coordinate plane. Here are some characteristics of linear equations that are important to understand:

• They contain no exponents or powers higher than one.
• Every term in a linear equation is either a number or a product of a number and a variable.
• They express a straight-line graph when plotted.
• Linear equations can have one or more variables, but the equation in its standard form is usually expressed as \( ax + b = c \).

To solve a linear equation, you need to isolate one of the variables, which means rearranging the equation so that one variable is alone on one side of the equation. This process often entails combining like terms and using simple arithmetic operations like addition and subtraction.

Integer Addition

Integer addition involves calculating the sum of whole numbers, which can be positive, negative, or zero. Understanding integer addition is crucial for completing various algebraic processes, including solving linear equations. In algebra, carefully handling signs during integer addition helps avoid mistakes.
Consider the equation from our problem, \( -10 + b = -17 \), to solve for \( b \), we performed an integer addition: \( -17 + 10 \). This calculation follows the rules of adding negative numbers:

• When adding two numbers with different signs, subtract the absolute value of each number and attach the sign of the larger number.
• If both numbers are negative, simply add their absolute values, and the result is negative.
• If both numbers are positive, add them normally, resulting in a positive number.

With practice, integer addition becomes a straightforward process, allowing you to manipulate numbers efficiently, even when solving more complex two-variable equations.

Two-variable Equations

Two-variable equations, like the one we encountered in this exercise, involve equations that contain two different unknowns. We are interested in finding the values of both variables that satisfy the equation simultaneously. This type of problem-solving is foundational in algebra, where students often encounter situations needing solutions for pairs of numbers.
When solving two-variable equations, such as \( a + b = -17 \), these strategies may help:

• **Choose a value for one variable:** Begin by assigning an arbitrary value to one of the variables. In our example, we set \( a = -10 \).
• **Substitute and solve for the other variable:** Substitute the chosen value into the equation and solve for the other variable—here, \( b = -17 + 10 \).
• **Verify your solution:** Check your answers by substituting both variables back into the original equation to ensure they satisfy it. This ensures the calculation is error-free.

Solving these types of problems enhances critical thinking and provides a basis for tackling more challenging algebraic concepts.