table of contents solving linear systems of equations - addition method recall that to solve the...

Table of Contents

Solving Linear Systems of Equations - Addition Method

• Recall that to solve the linear system of equations in two variables ... 222

111

cybxa

cybxa

we need to find the value of x and y that satisfies both equations. In this presentation the Addition Method will be demonstrated.

• Example 1:

2yx

4yx

Solve the following system

of equations ...

Table of ContentsSlide 2


2yx

4yx

• Label the equations as

# 1 and # 2. 2#

1#

• Equation # 2 states that x - y has the same value as - 2. Since we can add the same value to both sides of an equation to produce an equivalent equation, we proceed as follows.



• Start with equation # 1 ... 4yx • Add x - y (the left hand side of equation # 2) to the left hand side ...

yx 2

• Then add - 2 (the right hand side of equation # 2) to the right hand side ...

• The result is ...

60x2



60x2 • Solve this equation for x ...

3x • Now use either equation # 1 or equation # 2 to find the value of y. Using equation # 1 ...

4yx 4y3 1y

• The solution to the system is (3, 1), or x = 3 y = 1



213 • Since equation # 1 was used in the last step, check by substituting the solution values into equation # 2 ... 22

2yx

• Notice that in this system the coefficients of the y variables were the same except for sign (+ 1 and - 1).

• This is the form that a system must have right before the addition step. When the equations are added, one variable is eliminated, and the result is one equation with one unknown, which is easily solved for.

2yx

4yx



• Example 2:Solve the following system of equations ...

• Notice that neither variable meets the condition of coefficients being the same except for sign. This must be accomplished before proceeding.

3y2x7

8y4x3

2#

1#

• Multiplying equation # 2 by + 2 yields ... 6y4x14

8y4x3



6y4x14

8y4x3

• Now the coefficients on y

are the same, except for sign ...

• Addition of the equations yields ...

2x17

17

2x



• Substitute the value for x into equation # 1 (either equation could be used at this point) ...

17

2x

8y4x3

8y417

23

and solve for y ... 136y686

34

65y • The solution to the system is ...

17

2x

34

65y



• Example 3:Solve the following system of equations ...

• Neither variable meets the condition of coefficients being the same except for sign. The equations can be multiplied by constants to achieve this goal for either variable.

2y5x7

9y12x3

2#

1#

• The coefficients of x are 3 and 7. The lcm of 3 and 7 is 21, so multiply each equation by a convenient value to get coefficients of 21, opposite in sign.



9y12x3 • Multiply equation # 1 by 7 ...

63y84x21 and equation # 2 by - 3 ... 2y5x7

6y15x21

• Adding the two equations yields ...

63y84x21 6y15x21

57y99



33

19y • The solution for y is ...

• Substituting this value for y in equation # 2 (either equation could be used here) yields ... 33

23x

and the solution is ...

33

23x

33

19y



• Summary:1) Multiply the equations by constants so that one of the variables will have the same coefficients, opposite in sign.

2) Add the left sides and then the right sides of the two equations to yield one equation in one variable, for which we can solve.

4) To check the solutions, substitute both values into the equation that was not used in step 3.

3) Substitute the given value for the variable into either equation and solve for the other variable.

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