Solving Equations with Integers: A Comprehensive Approach

Solving equations with integers is similar to solving equations with real numbers, but it involves working with whole numbers, both positive and negative. Whether you're dealing with simple one-step equations or more complex equations, the general approach remains the same. Here's a comprehensive guide on how to solve equations with integers:

Step 1: Understand the Equation

• Carefully read and understand the equation, identifying the variable (typically represented by a letter, such as $x$), constants, coefficients, and operations (addition, subtraction, multiplication, division, etc.).

Step 2: Isolate the Variable

• Your goal is to isolate the variable on one side of the equation. To do this, perform operations on both sides of the equation to simplify and isolate the variable.

Step 3: Perform Operations

• Perform inverse operations to isolate the variable. If an operation is applied to one side of the equation, apply the same operation to the other side to maintain equality. Common inverse operations include:
  • Addition and subtraction (adding or subtracting the same value to both sides)
  • Multiplication and division (multiplying or dividing both sides by the same value)

Step 4: Simplify

• Simplify the equation as you perform operations. Combine like terms and follow the order of operations (PEMDAS/BODMAS) when necessary.

Step 5: Check for Extraneous Solutions (if applicable)

• After simplifying the equation, check for extraneous solutions if you've applied operations like taking square roots or other non-linear functions. Extraneous solutions are those that do not make the original equation true when substituted back in.

Step 6: Solve for the Variable

• Continue isolating the variable until it's on one side of the equation by itself. This is the solution to the equation.

Step 7: Express the Solution

• Express the solution as an integer or a set of integers, depending on the equation. The solution may be a single number, a range, or a set of numbers that satisfy the equation.

Step 8: Check the Solution

• Substitute the solution back into the original equation to verify that it makes the equation true. If it does, the solution is confirmed as correct.

Step 9: Finalize the Solution

• Review your work to ensure the equation has been solved accurately. Be aware of any special cases or restrictions that may apply to the variable.

Step 10: Express the Solution in Plain Language (if necessary)

• If you need to explain the solution in everyday language or in the context of a problem, do so for clarity.

Here's an example of solving a simple linear equation with integers:

Example: Solve the equation $-3x + 7 = 10$.

Step 1: Understand the Equation.

• Variable: $x$
• Constants: 7 and 10
• Coefficient: -3
• Operations: Addition and subtraction

Step 2: Isolate the Variable.

• $-3x + 7 - 7 = 10 - 7$

Step 3: Perform Operations.

• $-3x = 3$

Step 4: Simplify.

• $x = -1$

Step 5: Check for Extraneous Solutions.

• Not applicable in this case.

Step 6: Solve for the Variable.

• Variable is already isolated.

Step 7: Express the Solution.

• $x = -1$

Step 8: Check the Solution.

• Plug $x = -1$ back into the original equation: $-3(-1) + 7 = 10$, which is true.

Step 9: Finalize the Solution.

• Review the work to ensure accuracy.

Step 10: Express the Solution in Plain Language.

• The solution is $x = -1$, which means that when $x$ is -1, the equation $-3x + 7 = 10$ is true.

x + 3 - 3 = 7 - 3

x = 4

2x - 5 + 5 = 11 + 5

2x = 16

2x / 2 = 16 / 2

x = 8