7.5 Solving Systems of Linear Equations by Elimination

1. Solve linear systems by elimination. 2. Multiply when using the elimination method. 3.Use an alternative method to find the second value in a solution. 4. Solve special systems by elimination. 7.5

Objective 1 Solve linear systems by elimination.

Elimination Method An algebraic method that depends on the addition property of equality can also be used to solve systems. Adding the same quantity to each side of an equation results in equal sums: If A = B, then A + C = B + C. We can take this addition a step further. Adding equal quantities, rather than the same quantity, to each side of an equation also results in equal sums: If A = B and C = D, then A + C = B + D. The elimination method uses the addition property of equality to solve systems of equations.

Solve the system by the elimination method. Classroom Example 1 The solution set is {(2, –1)}. To find the corresponding y-value, substitute 2 for x in either of the two equations of the system. Using the Elimination Method

Solving a Linear System by Elimination Step 1 Write both equations in standard form Ax + By = C. Step 2 Transform the equations as needed so that the coefficients of one pair of variable terms are opposites. Multiply one or both equations by appropriate numbers so that the sum of the coefficients of either the x- or the y- terms is 0. Step 3 Add the new equations to eliminate a variable. The sum should be an equation with just one variable.

Solving a Linear System by Elimination Step 4 Solve the equation from Step 3 for the remaining variable. Step 5 Find the other value. Substitute the result from Step 4 into either of the original equations, and solve for the other variable. Step 6 Check the values in both of the original equations. Then write the solution set as a set containing an ordered pair.

Solve the system. Write both equations in standard form. Eliminate y. Classroom Example 2 The solution set is {(4, –2)}. Find the value of y. Using the Elimination Method

Objective 2 Multiply when using the elimination method.

Multiply when using elimination. Sometimes we need to multiply each side of one or both equations in a system by a number before adding will eliminate a variable.

Solve the system. Multiply each equation by a suitable number so that the coefficients of one of the two variables are opposites. Multiply equation (1) by 2 and equation (2) by 5. Classroom Example 3 Eliminate y. Using the Elimination Method

Solve the system. Find the value of y by substituting 2 for x in either equation (1) or equation (2). Classroom Example 3 Check that the solution set of the system is {(–2, 2)}. Using the Elimination Method (cont.)

Multiply when using elimination. In the previous example, we eliminated the variable y. Alternatively, we could multiply each equation of the system by a suitable number so that the variable x is eliminated. CAUTION! When using the elimination method, remember to multiply both sides of an equation by the same nonzero number.

Objective 3 Use an alternative method to find the second value in a solution.

Finding the Second Value Sometimes it is easier to find the value of the second variable in a solution using the elimination method twice.

Solve the system. Write each equation in standard form. Eliminate y by multiplying equation (1) by 2 and equation (2) by 3. Classroom Example 4 To solve for y, start over with the original equations and eliminate x. Finding the Second Value Using an Alternative Method

Solve the system. Each equation in standard form: Eliminate x by multiplying equation (1) by 3 and equation (2) by –2. Classroom Example 4 The solution set is Finding the Second Value Using an Alternative Method (cont.)

Objective 4 Solve special systems by elimination.

Solve each system by the elimination method. a. Multiply equation (1) by –2. The false statement 0 = 19 indicates that the given system has solution set Classroom Example 5 Solving Special Systems Using the Elimination Method

Solve each system by the elimination method. b. Multiply equation (1) by 2. A true statement occurs when the equations are equivalent. The solution set is {(x, y) | 2x + 5y = 1}. Classroom Example 5 Solving Special Systems Using the Elimination Method (cont.)