Solving Systems of Equations by Elimination (Addition) Section 4.3 Solving Systems of Equations by Elimination (Addition)

Objectives Solve systems of linear equations by the elimination method. Use multiplication to eliminate a variable. Use the elimination method twice to solve a system. Use elimination to identify inconsistent systems and dependent equations. Determine the most efficient method to use to solve a linear system.

Objective 1: Solve Systems of Linear Equations by the Elimination Method In the first step of the substitution method for solving a system of equations, we solve one of the equations for one of the variables. At times, this can be difficult, especially if none of the variables has a coefficient of 1 or −1. This is the case for the system Solving either equation for x or y involves working with cumbersome fractions. Fortunately, we can solve systems like this one using an easier algebraic method called the elimination or the addition method.

Objective 1: Solve Systems of Linear Equations by the Elimination Method The elimination method for solving a system is based on the addition property of equality: When equal quantities are added to both sides of an equation, the results are equal. In symbols, if A = B and C = D, then adding the left sides and the right sides of these equations, we have A + C = B + D. This procedure is called adding the equations.

EXAMPLE 1 Solve the system:

Objective 1: Solve Systems of Linear Equations by the Elimination Method To solve a system of equations in x and y by the elimination method, follow these steps. 1. Write both equations of the system in standard Ax + By = C form. 2. If necessary, multiply one or both of the equations by a nonzero number chosen to make the coefficients of x (or the coefficients of y) opposites. 3. Add the equations to eliminate the terms involving x (or y). 4. Solve the equation resulting from step 3. 5. Find the value of the remaining variable by substituting the solution found in step 4 into any equation containing both variables. Or, repeat steps 2–4 to eliminate the other variable. 6. Check the proposed solution in each equation of the original system. Write the solution as an ordered pair.

Objective 2: Use Multiplication to Eliminate a Variable In Example 1, the coefficients of the terms 5y in the first equation and −5y in the second equation were opposites. When we added the equations, the variable y was eliminated. For many systems, however, we are not able to immediately eliminate a variable by adding. In such cases, we use the multiplication property of equality to create coefficients of x or y that are opposites.

EXAMPLE 2 Solve the system:

Objective 3: Use the Elimination Method Twice to Solve a System Sometimes it is easier to find the value of the second variable of a solution by using elimination a second time.

EXAMPLE 5 Solve the system:

We can also solve these systems using the elimination method. Objective 4: Use Elimination to Identify Inconsistent Systems and Dependent Equations We have solved inconsistent systems and systems of dependent equations by substitution and by graphing. We can also solve these systems using the elimination method.

EXAMPLE 6 Solve the system:

Objective 5: Determine the Most Efficient Method to Use to Solve a Linear System If no method is specified for solving a particular linear system, the following guidelines can be helpful in determining whether to use graphing, substitution, or elimination. 1. If you want to show trends and see the point that the two graphs have in common, then use the graphing method. However, this method is not exact and can be lengthy. 2. If one of the equations is solved for one of the variables, or easily solved for one of the variables, use the substitution method. 3. If both equations are in standard Ax + By = C form, and no variable has a coefficient of 1 or −1, use the elimination method. 4. If the coefficient of one of the variables is 1 or −1, you have a choice. You can write each equation in standard (Ax + By = C) form and use elimination, or you can solve for the variable with coefficient 1 or −1 and use substitution.

Objective 5: Determine the Most Efficient Method to Use to Solve a Linear System Here are some examples of suggested approaches:

Objective 5: Determine the Most Efficient Method to Use to Solve a Linear System Each method that we use to solve systems of equations has advantages and disadvantages.