Problem 81 - Introductory and Intermediate Algebra for College Students 4th

Question

Write an equation with a negative solution that can be solved by adding 100 to both sides.

Step-by-Step Solution

Answer

An example of an equation with a negative solution that can be solved by adding 100 on both sides is -50 = -150.

Step 1: Choose a Negative Number for x

Select a negative number for x. For instance, choose x = -50.

Step 2: Define the Variable a

Given the equation x = a, we need a to be 100 less than x. So, apply this condition to the value chosen for x to calculate a. In this case, a will be (-50) - 100 = -150.

Step 3: Form the Equation

With the values of x and a determined, the equation x = a can now be written as: -50 = -150.

Step 4: Solve the Equation

Add 100 on both sides of the equation, which gives: -50 + 100 = -150 + 100. This simplifies to 50 = -50, and helps to verify the correctness of the original equation.

Key Concepts

Negative Solutions

Understanding negative solutions is an integral part of solving algebraic equations. A 'negative solution' refers to the value of the variable that makes the equation true, and this value is less than zero. In algebra, negative numbers are represented with a minus sign ( - ) before the number. When an equation is set up to find a negative solution, it means that the process of solving it will result in a negative number.

For example, if you have an equation such as \( x + 5 = 0 \) and you subtract 5 from both sides, you'll get \( x = -5 \). The solution here is \( x = -5 \) which is a negative number. It's crucial to understand that negative numbers follow the same arithmetic rules as positive numbers, just with a different sign. When dealing with negative solutions, always take care of the signs during each operation to avoid errors.

Adding Integers

Adding integers is a basic mathematical skill that involves combining two or more whole numbers (positive, negative, or zero). The rules for adding integers are straightforward but important to grasp for solving algebraic equations. When the signs are the same, you add the numbers and keep the same sign. For instance, \( (-3) + (-2) = -5 \), because both integers are negative.

When the signs are different, you subtract the smaller number from the larger one and take the sign of the number with the larger absolute value. For example, \( 7 + (-3) = 4 \) and \( (-7) + 3 = -4 \). The process of adding integers is vital when solving equations, especially when the goal is to isolate the variable to one side of the equation.

Equation Solving Steps

Solving an algebraic equation typically involves a series of steps that lead to finding the value of the variable. There's usually a starting point, often the original form of the equation, and through manipulation and application of algebraic rules, the equation is simplified to reveal the variable's value.

Basic equation solving steps include:

• Identifying the variable that needs to be solved for.
• Performing the same operation on both sides of the equation to maintain equality.
• Isolating the variable on one side of the equation usually through addition, subtraction, multiplication, or division.
• Simplifying the equation at each step to make the variable more apparent.
• Checking the solution by substituting it back into the original equation to verify if it satisfies the equation.

Once mastered, these steps make solving even complex equations a more manageable task.