1. Solving Linear Systems of Equations - Addition Method Recall that to solve the linear system of equations in two variables... we need to find the values of x and y that satisfy both equations. In this presentation the Addition Method will be demonstrated. Example 1: Solve the following system of equations...

2. Label the equations as # 1 and # 2. 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.

3. Start with equation # 1... Add x - y (the left hand side of equation # 2) to the left hand side... Then add 2 (the same value as x - y) to the right hand side... The result is...

4. Solve this equation for x... Now use either equation # 1 or equation # 2 to find the value of y. Using equation # 1... The solution to the system is (3, 1), or x = 3 y = 1

5. Since equation # 1 was used in the last step, check by substituting the solution values into equation # 2... x = 3 y = 1

6. 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.

7. 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. Multiplying equation # 2 by + 2 yields...

8. Now the coefficients on y are the same, except for sign... Addition of the equations yields...

9. Substitute the value for x into equation # 1 (either equation could be used at this point)... and solve for y... The solution to the system is...

10. Example 3: Solve the following system of equations... Note that neither equation is in ax + by = c form. Get both equations in this form before proceeding.

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

12. Multiply equation # 1 by 7... and equation # 2 by - 3... Adding the two equations yields...

13. The solution for y is... Substituting this value for y in equation # 2 (either equation could be used here) yields... and the solution is...

14. Summary: 2) Multiply the equations by constants so that one of the variables will have the same coefficients, opposite in sign. 3) 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. 5) Write the ordered pair solution ( x, y ) 4) Substitute the given value for the variable into either equation and solve for the other variable. 1) Write both equations in ax + by = c form.