Solve The System Of Equations By The Addition Method

The addition method, also known as the elimination method, stands as a powerful technique for solving systems of linear equations. Its beauty lies in its simplicity: manipulate the equations so that when added together, one variable vanishes, leaving you with a single equation in one variable, easily solvable. This method is particularly effective when coefficients of one variable are opposites or can be easily made into opposites.

Understanding Systems of Linear Equations

Before diving into the addition method, it’s crucial to understand what a system of linear equations represents. A system of linear equations is a collection of two or more linear equations involving the same variables. A solution to the system is a set of values for the variables that satisfies all equations simultaneously. Geometrically, each linear equation represents a line. The solution to a system of two linear equations in two variables represents the point of intersection of the two lines. If the lines are parallel, there is no solution. If the lines are the same, there are infinitely many solutions.

The Addition Method: A Step-by-Step Guide

Here’s a detailed breakdown of how to solve a system of equations using the addition method:

1. Align the Equations: Ensure that like terms (terms with the same variable) are aligned vertically. Write the equations one above the other, with the x-terms in one column, the y-terms in another, and the constant terms on the right-hand side of the equals sign. If necessary, rearrange the terms within an equation to achieve this alignment. For instance, if you have the system:

3x + 2y = 7
y = 5 - x

You would rewrite the second equation as:

x + y = 5

So the aligned system becomes:

3x + 2y = 7
x + y = 5

2. Identify a Variable to Eliminate: Look for a variable whose coefficients are either opposites (e.g., 3 and -3) or can easily be made into opposites by multiplying one or both equations by a constant. The goal is to have the coefficients of one variable be additive inverses of each other.

Example: In the system:

 2x + y = 8
 x - y = 1

The coefficients of *y* are already opposites (+1 and -1).  This makes *y* an excellent candidate for elimination.

Example: In the system:

  x + 3y = 10
  2x + y = 5

 Neither *x* nor *y* has coefficients that are opposites. However, we could easily multiply the second equation by -3 to make the *y* coefficients opposites (+3 and -3).  Alternatively, we could multiply the first equation by -2 to make the *x* coefficients opposites (+1 and -2).

3. Multiply (if necessary): If the coefficients of the chosen variable are not opposites, multiply one or both equations by a suitable constant(s) to make them opposites. Remember that when you multiply an equation by a constant, you must multiply every term in the equation by that constant to maintain the equality.

Example: Consider the system:

  3x + 4y = 11
  x + 2y = 5

 To eliminate *x*, we could multiply the second equation by -3:

  3x + 4y = 11
  -3(x + 2y) = -3(5)

 This gives us:

  3x + 4y = 11
  -3x - 6y = -15

 Now the *x* coefficients are opposites (+3 and -3).

4. Add the Equations: Add the two equations together, term by term. The variable with opposite coefficients should cancel out, leaving you with a single equation in one variable.

Example: Using the system from the previous step:

  3x + 4y = 11
  -3x - 6y = -15

 Adding the equations gives:

  (3x - 3x) + (4y - 6y) = (11 - 15)
  0x - 2y = -4
  -2y = -4

5. Solve for the Remaining Variable: Solve the resulting equation for the remaining variable. This is usually a straightforward algebraic step.

Example: From the previous step, we have:

  -2y = -4

 Dividing both sides by -2 gives:

  y = 2

6. Substitute to Find the Other Variable: Substitute the value you found in the previous step back into either of the original equations (or any of the modified equations) to solve for the other variable. Choose the equation that looks easiest to work with.

Example: Using the original equation x + 2y = 5 and the value y = 2:

  x + 2(2) = 5
  x + 4 = 5
  x = 1

7. Check Your Solution: Substitute the values you found for both variables into both of the original equations to verify that they satisfy both equations simultaneously. This is crucial to catch any errors made during the process.

Example: We found x = 1 and y = 2. Let's check with the original system:

  3x + 4y = 11  =>  3(1) + 4(2) = 3 + 8 = 11 (Correct)
  x + 2y = 5   =>  1 + 2(2) = 1 + 4 = 5 (Correct)

Since the solution satisfies both equations, it is the correct solution.

8. Express the Solution: Write the solution as an ordered pair (x, y).

Example: The solution to the system is (1, 2).

Examples of the Addition Method in Action

Let's work through some more examples to solidify your understanding.

Example 1:

Solve the following system of equations:

x + y = 5 x - y = 1

Step 1: Align the Equations (Already aligned)

Step 2: Identify a Variable to Eliminate The y coefficients are already opposites.

Step 3: Multiply (if necessary) Not necessary in this case.

Step 4: Add the Equations

(x + x) + (y - y) = (5 + 1) 2x + 0 = 6 2x = 6

Step 5: Solve for the Remaining Variable

2x = 6 x = 3

Step 6: Substitute to Find the Other Variable Using the equation x + y = 5:

3 + y = 5 y = 2

Step 7: Check Your Solution

x + y = 5 => 3 + 2 = 5 (Correct) x - y = 1 => 3 - 2 = 1 (Correct)

Step 8: Express the Solution The solution is (3, 2).

Example 2:

Solve the following system of equations:

2x + 3y = 8 x - y = 1

Step 1: Align the Equations (Already aligned)

Step 2: Identify a Variable to Eliminate We can eliminate x by multiplying the second equation by -2, or eliminate y by multiplying the second equation by 3. Let's eliminate x.

Step 3: Multiply (if necessary) Multiply the second equation by -2:

-2(x - y) = -2(1) -2x + 2y = -2

Step 4: Add the Equations

(2x - 2x) + (3y + 2y) = (8 - 2) 0x + 5y = 6 5y = 6

Step 5: Solve for the Remaining Variable

5y = 6 y = 6/5

Step 6: Substitute to Find the Other Variable Using the equation x - y = 1:

x - (6/5) = 1 x = 1 + (6/5) x = 11/5

Step 7: Check Your Solution

2x + 3y = 8 => 2(11/5) + 3(6/5) = 22/5 + 18/5 = 40/5 = 8 (Correct) x - y = 1 => (11/5) - (6/5) = 5/5 = 1 (Correct)

Step 8: Express the Solution The solution is (11/5, 6/5).

Example 3:

Solve the following system of equations:

4x + 6y = 2 6x + 9y = 3

Step 1: Align the Equations (Already aligned)

Step 2: Identify a Variable to Eliminate To eliminate x, we can multiply the first equation by -3 and the second equation by 2.

Step 3: Multiply (if necessary)

-3(4x + 6y) = -3(2) => -12x - 18y = -6 2(6x + 9y) = 2(3) => 12x + 18y = 6

Step 4: Add the Equations

(-12x + 12x) + (-18y + 18y) = (-6 + 6) 0x + 0y = 0 0 = 0

Step 5: Solve for the Remaining Variable Since we ended up with 0 = 0, this indicates that the two equations are dependent, meaning they represent the same line. Therefore, there are infinitely many solutions. We can express the solution in terms of one variable. Let's solve the first equation for x:

4x + 6y = 2 4x = 2 - 6y x = (2 - 6y) / 4 x = (1 - 3y) / 2

Step 6: Substitute to Find the Other Variable Not applicable in this case, as we have infinitely many solutions.

Step 7: Check Your Solution Any value of y will yield a corresponding value of x that satisfies both equations.

Step 8: Express the Solution The solution can be expressed as ((1 - 3y) / 2, y), where y can be any real number. This represents the set of all points on the line.

When the Addition Method is Most Useful

The addition method shines when:

• Coefficients are Opposites: If the coefficients of one variable are already opposites, the addition method requires minimal effort.
• Easy Multiplication: If the coefficients of one variable can be easily made into opposites by multiplying one or both equations by a small integer, the addition method is often the quickest approach.
• Integer Coefficients: The addition method tends to be easier to apply when dealing with equations with integer coefficients, as it avoids fractions for as long as possible.

Comparison with Other Methods

While the addition method is powerful, it's not the only way to solve systems of linear equations. Two other common methods are:

• Substitution Method: In the substitution method, you solve one equation for one variable and substitute that expression into the other equation. This method is often preferred when one equation is already solved for one variable or when it's easy to isolate one variable.

• Graphing Method: The graphing method involves graphing each equation and finding the point of intersection. While visually appealing, this method is generally less accurate than algebraic methods, especially when the solutions are not integers.

The best method to use depends on the specific system of equations you're dealing with. Sometimes, one method is clearly more efficient than the others. With practice, you'll develop a sense of which method is most appropriate for a given problem.

Common Mistakes to Avoid

• Forgetting to Multiply All Terms: When multiplying an equation by a constant, make sure to multiply every term in the equation, including the constant term on the right-hand side.
• Incorrectly Adding Equations: Double-check your addition, especially when dealing with negative numbers.
• Substituting into the Same Equation: After solving for one variable, don't substitute its value back into the same equation you used to solve for it. This won't give you any new information. Substitute into one of the other original equations.
• Not Checking Your Solution: Always check your solution by substituting the values back into the original equations to ensure they are satisfied. This helps catch any arithmetic errors.

Conclusion

The addition method is a fundamental and versatile technique for solving systems of linear equations. By strategically manipulating the equations to eliminate one variable, you can reduce the problem to a simple equation in one variable, which is easily solved. With a solid understanding of the steps involved and diligent practice, you'll master this powerful tool and be well-equipped to tackle a wide range of linear equation problems. Remember to always check your solutions to ensure accuracy and to consider the other available methods for solving systems of equations to choose the most efficient approach.