1	MATRIX SOLUTIONS OF EQUATIONS ROW-ECHELON FORM (MODIFIED)
2	TRANSLATE TO A MATRIX For the system: 2x – y + 3z = 4 x + 2y - z = -3 4x + 3y + 2z = -5 … Enter the coefficients only. First row:
3	TRANSLATE TO A MATRIX For the system: 2x – y + 3z = 4 x + 2y - z = -3 4x + 3y + 2z = -5 … Enter the coefficients only. First row: What shows where the equals sign belongs?
4	TRANSLATE TO A MATRIX For the system: 2x – y + 3z = 4 x + 2y - z = -3 4x + 3y + 2z = -5 Second row:
5	TRANSLATE TO A MATRIX For the system: 2x – y + 3z = 4 x + 2y - z = -3 4x + 3y + 2z = -5 Third row:
6	What Can We Do? [Elementary Row Operations] Swap rows Multiply a row by a constant Multiply a row by a constant and add the result to another row Same as swapping equations Same as multiplying an equation by a constant Same as multiplying an equation by a constant and adding the result to another equation
7	How do we actually solve the system? 1. Find a pivot (among the coefficients): an element that divides into the other elements in the same column What are the possible pivots here? 1 in column 1, -1 in col 2, or -1 in col 3 1’s and -1’s are ideal choices because they avoid fractions. Choosing 1 or -1 in column 1 gets you closest to the formal row-echelon form
8	How do we actually solve the system? 1. Find a pivot 2. Move the pivot row to the top
9	How do we actually solve the system? 1. Find a pivot 2. Move the pivot row to the top 3. Use the pivot element to zero out every element below the pivot -- Copy the pivot row without change -- Add a multiple of the pivot row to each lower row: choose the multiple to zero out the entries below the pivot
10	How do we actually solve the system? 1. Find a pivot 2. Move the pivot row to the top 3. Use the pivot element to zero out every element below the pivot
11	How do we actually solve the system? 1. Find a pivot 2. Move the pivot row to the top 3. Use the pivot element to zero out every element below the pivot 4. Cover up the top row, and repeat the whole process for the rows below.
12	How do we actually solve the system? 1. Find a pivot 2. Move the pivot row to the top 3. Use the pivot element to zero out every element below the pivot 4. Cover up the top row, and repeat the whole process for the rows below.
13	How do we actually solve the system? 1. Find a pivot 2. Move the pivot row to the top 3. Use the pivot element to zero out every element below the pivot 4. Cover up the top row, and repeat the whole process for the rows below.
14	How do we actually solve the system? 1. Find a pivot 2. Move the pivot row to the top 3. Use the pivot element to zero out every element below the pivot 4. Cover up the top row, and repeat the process
15	How do we actually solve the system? It is sensible to reduce any rows with a common factor. Here, divide row 2 by -5
16	How do we actually solve the system? Finish by back-substitution When we get to the bottom, we are ready to find the solutions. Translate back to equations Solve the bottom equation first. In this case, it is already done
17	How do we actually solve the system? Finish by back-substitution Now substitute into the previous equation. Solve for the second variable
18	How do we actually solve the system? Finish by back-substitution Now substitute into the first equation. Solve for the last variable Sol: (4,-5,-3)