Math 141 Lecture Notes



Math 141 Lecture Notes

Section 2.3 Systems of Linear Equations: Undetermined and Overdetermined Systems

Solutions of Linear Equations

Example 1: x - 2y + z = -3

2x + y - 2z = 2

x + 3y - 3z = 5

Example 2: 3x - 9y + 6z = -12

x - 3y + 2z = - 4

2x - 6y + 4z = 8

If there is a row in the augmented matrix containing all zeros to the left of the vertical line and a nonzero entry to the right of the line, then the system of equations has no solution.

Theorem

a. If the number of equations is greater than or equal to the number of variables in a linear system, then one of the following is true:

i. The system has no solution.

ii. The system has exactly one solution.

iii. The system has infinitely many solutions.

b. If there are fewer equations than variables in a linear system, then the system either has no solution or it has infinitely many solutions.

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Example 3: x + 2y = 3

2x - 3y = -8

x - 4y = -9

Example 4: x + 2y + 4z = 2

x + y + 2z = 1

Example 5: Mr. and Mrs. Garcia have a total of $100,000 to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of 12%/year, while the bonds and the money market account pay 8% and 4% per year, respectively. They have stipulated that the amount invested in stocks should be equal to the sum of the amount invested in bonds and 3 times the amount invested in the money market account. How should the Garcias allocate their resources if they require an annual income of $10,000 from their investments?

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