Math 1410, Spring 2020
Systems of Equations
January 28, 2020
| Sean Fitzpatrick |
|---|
| University of Lethbridge |
 |
Recap
Warm-up
Find the distance from the point \(P=(1,2,3)\) to the line
Equations of lines and planes
Lines are given by equations like
Planes are given by equations like \(2x+5y-10z=15\text{.}\)
These equations are linear. Intersections (two lines, line-plane, two planes) involve satisfying multiple linear equations simultaneously.
Systems of Linear Equations
Terminiology
Definition:
An equation involving variables \(x_1,x_2,\ldots, x_n\) is linear if it only involves addition and subtraction of constant multiples of the variable, and constant terms.
A system of linear equations is a collection of one or more linear equations, all involving the same variables. (Although not all variables must appear in all equations.)
A solution to a system of linear equations is a set of values for the variables that satisfies all equations in the system simultaneously.
Examples
Equations like \(3x-4=7\text{,}\) \(\frac12 x - \sqrt{2}y = 10\text{,}\) and
Equations like \(y=x^2\text{,}\) \(x^2-4y^2+z=4\text{,}\) and \(5wxyz=11\) are not. We can multiply variables by constants, but not by each other.
A system of linear equations could look like:
or
A solution to the second system is \(x=2, y=2, z=1\text{.}\) (Why?)
Solving systems: elementary operations
Any system of linear equations can be solved using the following three operations:
Swap positions of two equations. (Change the order)
Multiply both sides of an equation by a nonzero constant.
Add a multiple of one equation to another.
Note that none of these affect the values that will satisfy each equation.
Other operations might be legal, but the above can be “systematized”.
Example: interection of two lines
Suppose we want to know if the lines
Example: intersection of two planes
The planes \(x-2y+3z=6\) and \(2x+5y-4z=-4\) are not parallel, so they must intersect (in a line). We want to find an equation of this line.
Every point on the line belongs to both planes, so it must satisfy both equations simultaneously. That means we are looking for all solutions to the system
Elementary row operations
The augmented matrix
If we're going to be solving lots of linear systems (and we are), then it gets tedious quickly to keep track of all the variables.
A “key idea”: write only the coefficients, and keep track of variables by position. We organize everything in a rectangular array, called a matrix.
Example:
System of equations:
Augmented matrix:
From matrix to system
It's important to be able to convert in both directions:
Example:
Write down the system (in variables \(x_1,x_2,x_3,x_4\)) corresponding to the augmented matrix
Elementary row operations
There are three elementary operations on the rows of a matrix, corresponding to the elementary operations on equations:
Swap any two rows: write \(R_i\leftrightarrow R_j\)
Multiply a row by a nonzero constant (rescale): write \(cR_j\to R_j\)
Add a multiple of one row to another: write \(R_i+cR_j \to R_i\)
Gaussian elimination
There is a standard algorithm for solving a system using row operations:
-
By swapping rows and/or rescaling, get a 1 in row 1, column 1.
(This is possible unless column 1 consists entirely of zeros.)
By adding multiples of row 1 to the other rows, create zeros in all other entries of column 1.
Move to row 2, column 2, and repeat (until you reach the last row or column).
Example
For the system below, write down the augmented matrix, and use Gaussian elimination to simplify.