1. 4.8 Using matrices to solve systems! 2 variable systems – by hand 3 or more variables – using calculator!

2. Writing systems using matrices To rewrite a system using matrices 1 st, set up the equations in consistent order (as if you were solving by elimination method) 2 nd, you need to write 3 matrices: A coefficient matrix A variable matrix (this will be a column matrix) A constant matrix (this will be a column matrix as well) For example, the system of equations: 3x + 5y = 19 2x – 7y = 13 Becomes the following matrix equation:

3. Note that if we multiply the equation back out, we will obtain the original system of equation To solve the equation, we need to get rid of the coefficient matrix Do this by multiplying each side on the LEFT by the inverse of the coefficient matrix!

4. Example 8-1a Write a matrix equation for the system of equations. Determine the coefficient, variable, and constant matrices. Write the matrix equation. Answer: A XB =

5. Example 8-1b Write a matrix equation for the system of equations. Answer:

6. Example 8-3a Use a matrix equation to solve the system of equations. The matrix equation when

7. Example 8-3b Step 1Find the inverse of the coefficient matrix. Step 2Multiply each side of the matrix equation by the inverse matrix. Multiply each side by A –1.

8. Example 8-3c Multiply matrices. Answer:The solution is (5, –4). Check this solution in the original equations.

9. Example 8-3d Use a matrix equation to solve the system of equations. Answer: (2, –4)

10. Example 8-4a Use a matrix equation to solve the system of equations. The matrix equation is when

11. Example 8-4b Find the inverse of the coefficient matrix. The determinant of the coefficient matrixis 0, so A –1 does not exist.

12. Example 8-4c Graph the system of equations. Since the lines are parallel, this system has no solution. The system is inconsistent. Answer: There is no solution of this system.

13. Example 8-4d Use a matrix equation to solve the system of equations. Answer: no solution