Systems Solved by Adding and Multiplying

In addition to graphing the results of equations and using the substitution method, you can also solve equations with a method that involves using multiplication and addition processes to eliminate variables from the equations. To use this approach, you first assess the system of equations to determine a value by which you can multiply one of the equations so that you can make one or another of its coefficients the additive inverse of the corresponding coefficient in the other equation. You then multiply by this value and carry out an addition operation to arrive at a new equation in which you have eliminated one of the variables. You repeat this process until you arrive at values for each of the variables in the system.

To see how this works, consider this system of equations:

4x - 4y = -1

- 4x + 2y = 0

When you assess this system of equations, you can see that the coefficients of x are additive inverses. Since your goal is to arrive at a new equation in which you eliminate one of the two variables, adding these two equations immediately provides you with a desired result. When you carry out the addition, your activity takes the following form:

The addition eliminates x as a variable and leaves you with the equation -2y = -1. To find the value of y, you need to eliminate the coefficient of y, and to accomplish this, you multiply the equation by :

The result of this activity is the value of y:

This, then, provides you with half of your goal. Now that you have the value of y, you can proceed with discovering the value of x. To discover the value of x, you bring forward the first equation from the system of equations:

4x - 4y = -1

Your goal this time around is to eliminate the coefficient of y. To accomplish this, you make use of the fact that you know the value of y. The coefficient of y in the equation is -4. You must multiply the equation that establishes the value of y by a number that generates the additive inverse of -4. To reach this goal, you multiply y = by 4. Your activity in this respect takes the following form:

The result is an equation that eliminates the y variable if you add it to the first equation in the system of equations. Here is the operation that accomplishes this task:

The result of the operation is an equation that isolates x. You can then multiply this equation by to arrive at the value of x. Here is the multiplication and the result:

Now you know the values of both x and y. The value of x is , and the value of y is .

To test the correctness of your calculations, you can substitute the values of x and y into the equations of the original system. Here is the substitution for the first equation:

Here is the substitution for the second equation:

Exercise Set 11.2 Solve the following systems of equations using the elimination method. Check your answers. 3a + 2b = 5 a + 4b = 10 a - 2b = 6 2a - 3b = 4 6a - 12b = 16 3a - 4b = 8 5a + 4b = 5 2a - 4b = 8 2a + 3b = 17 5a + 7b = 29