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# Systems of Linear Equations

A system of equations is a collection of 2 or more
equations each containing 1 or more

variables. A system of linear equations has equations of the form a_{1}x_{1}+a_{2}x_{2}+· ·
·+a_{n}x_{n} = b.

• A solution consists of values for the variables that
satisfy (are solutions of) each

equation in the system. Solving the system of equations requires finding the
solution

set, which is the set of all such solutions.

• A system of equations is called consistent if it has one or more solutions;
otherwise

it is called inconsistent.

• A system of equations is called independent if it has exactly one solution and
dependent

if it has more than one solution.

*** System of Two Linear Equations in Two Variables**

ax + by = h

cx + dy = k

with variables x and y, coefficients a, b, c and d, and constants h and k.

***Graphing Method**

The graph of each equation is a line. There are three possible scenarios for

the solution set.

1. __One Solution__ - lines intersect at a point (consistent, independent)

2. __No Solutions__ - lines are parallel (inconsistent)

3. __Infinitely Many Solutions__ - lines coincide (consistent, dependent)

***Substitution Method**

Pick one equation and solve for one variable in terms of the other. Substitute

result into the other equation and solve resulting linear equation in one
variable.

Substitute result back into result from first step and solve for the second

variable.

***Elimination Method**

Elimination by addition involves the replacement of systems of equations with

simpler equivalent systems, by performing appropriate operations, until
obtaining

a system with an obvious solution. This method is very important

because it is readily generalized to higher-order systems.

Operations that transform a system of equations into an equivalent system

include:

1. Interchanging two equations

2. Multiplying an equation by a nonzero constant

3. Adding a constant multiple of one equation to another

***Matrices**

The system of equations

ax + by = h

cx + dy = k

can be represented without writing the symbols for the variables as an augmented

matrix.

A matrix is a rectangular array of numbers written within
brackets. Each

number in a matrix is called an **element.** If a matrix has m rows and n

columns, it is called an m × n **matrix.** The element in the ith row and jth

column of matrix A is denoted a_{ij}.

Operations that transform an augmented matrix into a row-equivalent matrix

include:

1. Interchanging two rows (R_{i} <--> R_{j})

2. Multiplying a row by a nonzero constant (kR_{i} --> R_{i})

3. Adding a constant multiple of one row to another (kR_{j} + R_{i}
--> R_{i})

Note: The arrow means “replace”.

**Solving System:** The system of equations represented by an augmented

matrix can be solved by transforming it into one of the following forms where

where m, n, and p are real numbers.

Form 1 - The system has an obvious unique solution.

Form 2 - The system has infinitely many solutions.

Form 3 - The system has no solution.

***Solving Linear Systems of Any Size**

Any system of linear equations must have exactly one solution, no solution, or

infinitely many solutions.

***Gauss-Jordan Elimination**

1. Choose the leftmost nonzero column and use appropriate row operations

to get a 1 at the top.

2. Use multiples of the row containing the 1 from step 1 to get zeros in all

remaining places in the column containing this 1.

3. Repeat step 1 using the submatrix formed using only rows below the row

used in step 2.

4. Repeat step 2 with the entire matrix and continue until it is impossible

to go further.

If at any point, a row is obtained with all zeros to the left of the vertical
line,

and a nonzero to the right, stop because there is **no solution.**

If the number of leftmost 1’s in the resulting matrix is less than the number of

variables in the system and there are no contradictions, then the system has

**infinitely many solutions.**

***Systems of Linear Inequalities in Two Variables**

***Graphing**

• a vertical line divides a plane into **left **and **right half-planes**

• a non-vertical line divides a plane into **upper** and
**lower half-planes**

• For an inequality Ax + By < C, Ax + By ≤C, Ax + By > C, or

Ax + By ≥C with B ≠ 0, the graph is either the lower or upper halfplane

(but not both) determined by the line Ax + By = C.

**Steps to Graph Inequality in Two Variables
**1. Graph the line Ax + By = C as a dashed line if equality is not

included in the original statement (< or >), or as a solid line if

equality is included ( ≤or≥ ).

2. Choose a test point anywhere in the plane not on the line and

substitute the coordinates into the inequality. (Hint: A common

test point is (0,0) because it is easy to evaluate.)

3. If the inequality is satisfied by the point, the graph is the half-plane

that includes the test point, otherwise, the graph is the half-plane

that does not include the test point.

• For the inequality Ax < C or Ax > C, the graph is either the left or

right half-plane (but not both) determined by the line Ax = C.

***Solving
**• Graphical solution is to find the intersection of the graphs of all equations

in the system, called the

**solution region**or

**feasible region.**

•

**A corner point of**a solution region is the intersection of two boundary

lines.

• The inequalities x ≥0 and y ≥0, called

**nonnegative restrictions,**occur

frequently in applications involving systems in which x and y cannot

be negative.

• A solution region is

**bounded**if it can be enclosed with a circle.

• A solution region is

**unbounded**if it cannot be enclosed with a circle.