Problem 185 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve the system. The sum of two numbers is negative thirty. One number is five times the other. Find the numbers.

Step-by-Step Solution

Answer

The numbers are -25 and -5.

Step 1: Define the Variables

Let the first number be denoted as x and the second number be denoted as y.

Step 2: Set Up the Equations

Using the given information, we can set up the system of equations: 1. The sum of two numbers: \(x + y = -30\)2. One number is five times the other: \(x = 5y\)

Step 3: Substitute One Equation into the Other

Substitute \(x = 5y\) from the second equation into the first equation: \(5y + y = -30\)

Step 4: Solve for y

Combine like terms and solve for y: \(6y = -30\) Divide both sides by 6: \(y = -5\)

Step 5: Solve for x

Substitute \(y = -5\) back into \(x = 5y\): \(x = 5(-5)\) \(x = -25\)

Step 6: State the Final Answer

The two numbers are \(x = -25\) and \(y = -5\).

Key Concepts

algebraic equations

An algebraic equation is an equation that includes at least one variable. This variable can represent unknown numbers, which we aim to find. Algebraic equations often use symbols like x or y to represent these unknowns. In our given problem, the equations we are dealing with are:
x + y = -30
x = 5y
We are working with two equations involving the variables x and y. These equations help us understand how the two unknown numbers relate to each other. Algebraic equations are fundamental in various fields of mathematics and science because they allow us to represent and solve real-life problems mathematically.

substitution method

The substitution method is one of the ways to solve a system of equations. It involves solving one equation for one variable and substituting that solution into another equation. This effectively reduces the system to a single equation with one variable, making it easier to solve.
In our problem, we have:
1. x + y = -30
2. x = 5y
Step 1: From the second equation, we see that x is already expressed in terms of y: x = 5y.
Step 2: We substitute x = 5y into the first equation:
5y + y = -30
This gives us a new, simplified equation in terms of y:
6y = -30
Now, solving for y becomes straightforward. Once we find y, we can go back and find x using x = 5y.
The substitution method is powerful because it simplifies the problem step-by-step, making it an ideal choice for solving systems of equations with two variables.

solving equations step by step

Let's break down the process of solving equations step by step, as demonstrated in the exercise:

1. **Define the Variables**
First, identify the unknowns and express them as variables. In our case, let x and y represent the two numbers.

2. **Set Up the Equations**
Using the problem's information, set up the equations that describe the relationships between the variables:
x + y = -30
x = 5y

3. **Substitute One Equation into the Other**
Use one of the equations to express one variable in terms of the other and substitute this into the second equation. For our problem, substitute x = 5y into x + y = -30 to get:
5y + y = -30
6y = -30

4. **Solve for y**
Combine like terms and solve for y by dividing both sides by 6:
y = -5

5. **Solve for x**
Substitute the value of y back into one of the original equations to find x. Using x = 5y:
x = 5(-5)
x = -25

6. **State the Final Answer**
Finally, write down the solutions clearly:
The two numbers are x = -25 and y = -5.
Following these steps ensures that you methodically solve the system of equations, reducing errors and making complex problems manageable.