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10.1

Systems of Linear Equations:Two Equations Containing Two

Variables

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Asystem of equations is a collection of two or

more equations, each containing one or more

variables.

Asolution of a system of equations consists of

values for the variables that reduce each equation

of the system to a true statement.

When a system of equations has at least one

solution, it is said to be consistent; otherwise it is

called inconsistent.

To solve a system of equations means to find all

solutions of the system.

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An equation in n variables is said to be

linear if it is equivalent to an equation ofthe form

where are n distinct variables,

are constants, and at leastone of the as is not zero.

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If each equation in a system of equations

is linear, then we have a system of linear

equations.

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If the graph of the lines in a system of two linear

equations in two variables intersect, then the system

of equations has one solution, given by the point ofintersection. The system is consistent and the

equations are independent.

Solution

y

x

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If the graph of the lines in a system of two linear

equations in two variables are parallel, then the

system of equations has no solution, because thelines never intersect. The system is inconsistent.

x

y

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If the graph of the lines in a system of two linear

equations in two variables are coincident, then the

system of equations has infinitely many solutions,represented by the totality of points on the line. The

system is consistent and dependent.

x

y

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Two Algebraic Methods for Solving

a System

1. Method of substitution

2. Method of elimination

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STEP 1: Solve forx in (2)

Use Method ofSubstitution to solve:

(1)

(2)

addx

and subtract 2

on both sides

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STEP 2: Substitute forx in (1)

STEP 3: Solve fory

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STEP 4: Substitute y = -11/5 into result in

Step1.

Solution:

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STEP 5: Verify solution

in

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Rules for Obtaining an Equivalent

System of Equations

1. Interchange any two equations of the

system.

2. Multiply (or divide) each side of an

equation by the same nonzero constant.

3. Replace any equation in the system by

the sum (or difference) of that equation

and any other equation in the system.

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Multiply (2) by 2

Replace (2) by the sum of (1) and (2)

Equation (2) has no solution. System is

inconsistent.

Use Method of Elimination

to solve:

(1)

(2)

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Multiply (2) by 2

Replace (2) by the sum of (1) and (2)

The original system is equivalent to a system containing

one equation. The equations are dependent.

Use Method of Elimination

to solve:

(1)

(2)

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This means any values x and y, for which 2x -y =4

represent a solution of the system.

Thus there is infinitely many solutions and they can be

written as

or equivalently