Subtraction of Equations

Subtracting equations involves finding the difference between two equations by subtracting one from the other. This process can be useful in solving systems of equations, where the goal is to eliminate a variable. In subtraction of equations, we deal with:

Coefficients: These are constant values like -1, 4, etc.

Variables: These are unknown quantities like x, y, z, etc.

Operators: For subtraction, the operator is the minus (-) symbol.

When subtracting equations, students should follow these rules:

Align terms: Make sure that all like terms are aligned vertically.

Subtract: Subtract the coefficients of the aligned terms and simplify the result.

Check solution: After subtracting, solve the resulting equation for the variable of interest.

The following are methods for subtracting equations:

Method 1: Elimination Method

To apply the elimination method for subtraction of equations, follow these steps.

Step 1: Arrange both equations in standard form with like terms aligned.

Step 2: Subtract one equation from the other to eliminate a variable.

Step 3: Solve the resulting equation.

Example: Subtract the equations 3x + 2y = 7 2x + 2y = 5

Step 1: Align the equations: 3x + 2y = 7 2x + 2y = 5

Step 2: Subtract the second equation from the first: (3x + 2y) - (2x + 2y) = 7 - 5

Step 3: Simplify and solve: x = 2 Answer: x = 2

Method 2: Substitution Method

In the substitution method, solve one equation for one variable and substitute into the other equation to eliminate the variable.

Example: Solve the equations x + y = 6 x - y = 2

Solution: Solve the first equation for x: x = 6 - y

Substitute in the second equation: (6 - y) - y = 2 6 - 2y = 2 2y = 4 y = 2

Substitute back: x = 6 - 2 = 4

Therefore, x = 4 and y = 2

Subtraction of equations has the following properties:

• Subtraction is not commutative In subtraction, changing the order of the terms changes the result, i.e., A - B ≠ B - A

• Subtraction is not associative Unlike addition, we cannot regroup in subtraction. When three or more equations are involved, changing the grouping changes the result. (A − B) − C ≠ A − (B − C)

• Subtraction is the addition of the opposite sign Subtracting an equation is the same as adding its opposite, so to make calculations easier, you can convert subtraction into addition by changing the signs of the second equation. A − B = A + (−B)

• Subtracting zero from an equation leaves the equation as is Subtracting zero from any equation results in the same equation: A - 0 = A

Tips and tricks for effectively dealing with subtraction of equations include:

Tip 1: Always check alignment of like terms before subtracting.

Tip 2: If two equations have identical terms on one side, subtract them out to simplify calculations.

Tip 3: Use the elimination method when a variable can be easily eliminated by subtraction.

Subtraction in equations can be challenging, leading to common mistakes. Awareness of these errors can help students avoid them.