1. Ch. 7 – Matrices and Systems of Equations 7.3 – Multivariable Linear Systems

2. Systems of 3 equations  Here is an example of a system of 3 linear equations:  Can do elimination to solve, but must do it a lot  Here is an example of the same system of 3 linear equations in Row-Echelon Form (REF):  REF  equations are in a stair-step pattern, leading coefficient is 1  It’s much easier to solve!  Just plug in 2 for z to find y, then plug that in to find x!  We want multivariable equations to look like this!

3. REF it!  Ex 1: Solve this system of equations.  To solve, we will do several eliminations to get the equations in REF!  To do that, we need our stair-step pattern and leading coefficients of 1!  ADD and REPLACE!  Step 1: First equation must be led by x - CHECK!  Step 2: Second equation must be led by y  Must get rid of x, so add equations 1 and 2  The new equation is y + 3z = 5, so replace it for the second equation to get…  Now 2 nd equation is led by y!

4. REF it!  (cont’d): Solve this system of equations:  Step 3: Third equation must be led by z  Must get rid of x, so add -2 (first equation) and second equation  Must get rid of y, so add 2 nd and 3 rd equations  Back-substitute to get y = -1 and x = 1, so we get…  … (1, -1, 2)

5. REF it!  Ex 2: Solve this system of equations. Divide 1 st equation by 2… Add -2(E2) and E3 and replace for E2… Add -4(E1) and E3 and replace for E3…

6. REF it!  Ex 2: Cont’d. Add 5(E2) and 3(E3) and replace for E3… Divide E2 and E3 down to REF…

7. REF it!  Ex 2: Cont’d.  Now back-substitute…  Answer: ( ½, -3/2, 1 )

8.  Ex 3: Solve this system of equations.  You can always rewrite the system as an augmented matrix (3x4) of coefficients – it will save you lead!  Just get 1’s in the main diagonal! Write as a 3x4 matrix… Add E1 and E2 and replace for E2… Add -2(E1) and E3 and replace for E3…

9.  Ex 3: cont’d. Add -4(E2) and E3 and replace for E3… Divide E3 by -1 to get…

10.  Ex 3: Cont’d.  Now back-substitute…  Answer: ( 69, 101, 46 )

11.  Ex 4: Solve this system of equations. Write as a 3x4 array… Add -2(E1) and E2 and replace for E2… Add E1 and E3 and replace for E3…

12.  Ex 4: Solve this system of equations.  We’re left with 0 = 0 in the 3 rd equation. Since this equation is true, we have an infinite number of solutions… Add 2(E2) and E3 and replace for E3…

13.  Ex (cont’d):  If the answer is infinite solutions, we must find a generic triple that solves the system:  Step 1: Let z = a.  Step 2: Back-substitute to get expressions for x and y in terms of a  Answer: ( 5/2 – ½a, 4a – 1, a )