Problem 31 - Elementary Algebra

Question

In the following exercises, solve each number word problem. The sum of two numbers is -316. One number is 94 less than the other. Find the numbers.

Step-by-Step Solution

Answer

The numbers are -111 and -205.

Step 1: Define Variables

Let's define the two unknown numbers. Let the first number be denoted as \( x \) and the second number as \( y \).

Step 2: Set Up Equations

We have two pieces of information: 1) The sum of the numbers is -316. \[ x + y = -316 \] 2) One number is 94 less than the other. \[ y = x - 94 \]

Step 3: Substitute and Solve for x

Substitute the second equation into the first equation: \[ x + (x - 94) = -316 \] Simplify and solve for \( x \): \[ 2x - 94 = -316 \] \[ 2x = -222 \] \[ x = -111 \]

Step 4: Solve for y

Use the value of \( x \) to find \( y \): \[ y = x - 94 \] \[ y = -111 - 94 \] \[ y = -205 \]

Step 5: Verify the Solution

Verify by checking the sum: \[ -111 + (-205) = -316 \] Both conditions are met.

Key Concepts

Solving Equations

In algebra, solving equations is like solving a puzzle. The goal is to find the unknown numbers that make the equation true. For the problem given, we were told that the sum of two numbers is -316 and one number is 94 less than the other.
To solve this, we followed several steps:
- We defined variables to represent the unknown numbers: let’s say the first number is \( x \) and the second number is \( y \).
- We used the information given to set up equations. The first piece of information told us \( x + y = -316 \). The second told us that one number is 94 less than the other, so: \( y = x - 94 \).
Solving the equations is like using clues. By substituting the second equation into the first, we simplified to solve for \( x \). Always check your work to ensure the solution fits all parts of the problem!

System of Linear Equations

A system of linear equations is a set of two or more equations with the same variables. In our problem, we had the system:
\[ \begin{cases} x + y = -316 \ y = x - 94 \end{cases} \]
The goal is to find values for \( x \) and \( y \) that satisfy both equations at the same time.
There are several methods to solve a system of linear equations:

• Graphing
• Substitution
• Elimination

For this problem, we used substitution because it directly incorporates one equation into the other. This method is particularly handy when one equation is already solved for one variable.

Variable Substitution

Variable substitution is a technique where you solve one equation for one variable and then substitute that expression into another equation. This method helps to simplify complex problems.
For our exercise, we had:
\[ x + y = -316 \] \[ y = x - 94 \]
By substituting \( y = x - 94 \) into \( x + y = -316 \):

• You replace \( y \) in the first equation with \( x - 94 \)
• The equation becomes \( x + (x - 94) = -316 \)

This simplifies to:
\[ 2x - 94 = -316 \]
Solving this gives us \( x = -111 \). Then, substitute \( x \) back to find \( y \):
\( y = -111 - 94 = -205 \).
Substitution is a powerful tool that transforms a system of equations into easier steps, which leads us to the final answer.