Section 2.1 and 2.2

A System of Linear Equations is when we have two or more linear equations working together. The solution to a linear system is a set of values to the variables such that all the equations are simultaneously satisfied. There are two common techniques to find the solutions of system of linear equations, one way is called substitution, and the other way is elimination. Let's started with considering a system of only two linear equations with two variables.

Eg. Solve the following equations.

Review of May 26:

Does the solution always exist? Do you always have unique solution?

Setting up linear system for real life problems

A bakery sells muffins, scones, and croissants. Each batch of 6 muffins uses 2 cups of sugar and 3 cups of flour. Muffins sell for $2.50 each batch. Each batch of 10 scones uses 2 cups of sugar and 5 cups of flour. Scones sell for $2.00 each batch. Each batch of 12 croissants uses 1 cup of sugar and 4 cups of flour. Croissants sell at $1.50 each batch. The bakery has 17 cups of sugar and 37 cups of flour, and for one day the total revenue is $169. Determine how many muffins, scones, and croissants were sold. Set up, do not solve.

Augmented Matrices and Gauss-Jordan elimination Method

An augmented matrix is in reduced echelon form if:

RREF: This will help us to find the solution easily

1. All rows consisting entirely of zeros are grouped at the bottom of the matrix.2. The leftmost nonzero number in each row is 1. This element is called the ”leading 1of the row”.3. The leading 1 of a row is to the right of the leading 1 in the previous row.4. All entries directly above and below a leading 1 are zeros.

Row operations:

1. Interchange any two rows2. Replace any row by multiplying a nonzero constant to itself3. Replace any row by the sum of that row and a constant multiple of any other row

How to use your calculator?

Step 1: press 2nd, then press x^-1 to go to the matrix functionStep 2: To edit a matrix, use direction keys to choose Edit on the top row. Step 3: Edit by use direction keys.Step 4: To find RREF of a matrix, use direction keys to choose Math on the top row.Step 5: Find your saved matrix by going to the matrix menu again.

Solve the following systems

2x+4y+6z = 223x+8y+5z = 27-x+y+2z = 2

8z+3x-2y = 92y+z = 3+2xX+2y-3z-8 = 0

2y+3z = 73x+6y-12z = -35x-2y+2z = 7

End