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Section 3.2 Solving Systems of Equations Algebraically. Use algebraic computation to get precise solutions The Substitution Method (One Equation into Another) The Elimination Method (Adding Equations) How to identify Consistent Systems (one solution – lines cross)

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Section 3.2 Solving Systems of Equations Algebraically


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section 3 2 solving systems of equations algebraically
Section 3.2 Solving Systems of Equations Algebraically
  • Use algebraic computation to get precise solutions
  • The Substitution Method (One Equation into Another)
  • The Elimination Method (Adding Equations)
  • How to identify
    • Consistent Systems (one solution – lines cross)
    • Inconsistent Systems (no solution – parallel lines)
    • Dependent Systems (infinitely many solutions – same line)
  • Comparing the Methods
definition
Definition

Simultaneous Linear Equations

Consider the pair of equations together

4x + y = 10

-2x + 3y = -12

Each line has infinitely many pairs (x, y) that satisfy it.

But taken together, only one pair (3, -2) satisfies both.

Finding this pair is called solving the system.

In 3.1, you learned to solve a system of two equations in two variables by graphing (approximation).

In this section you will learn two ways to solve linear systems algebraically (precision).

if a b and b c then a c substitution method example
if a=b and b=c then a=cSubstitution Method - Example
  • You can pick either variable to start – you will get the same (x,y) solution. Itmay take some work to isolate a variable:
  • Solve for (A)’s yor Solve for (A)’s x
if a b then ca cb elimination method even multiples
if a=b then ca=cbElimination Method – Even Multiples
  • When one variable term is an evenmultiple of its matching variable term,you only have to multiply one equation.
  • Let’s eliminate y: Multiply (A) by -2
elimination method no multiples
Elimination Method – No Multiples
  • When no variable terms are even multiples, you have to multiply bothequations by different numbers.
  • Let’s eliminate y: Multiply (A) by 7 and (B) by -3
0 4 untrue inconsistent systems how can you tell
0 = 4 untrueInconsistent Systems - how can you tell?
  • An inconsistent system has no solutions.

(parallel lines)

Substitution Technique Elimination Technique

0 0 or n n dependent systems how can you tell
0 = 0 or n = n Dependent Systems – how can you tell?
  • A dependent system hasinfinitely many solutions.

(it’s the same line!)Substitution Technique Elimination Technique

application problem board work
Application Problem (board work)

TEMPORARY HELP

A law firm had to hire several

workers to help finish a large

project. From the billing records

shown in Illustration 3,

determine the daily fee charged

by the employment agency for a

clerk-typist and for a computer

programmer.

The daily fees:

$105 for a clerk-typist

$185 for a programmer

geometry example board work
Geometry Example (Board work)

TRAFFIC SIGNAL

In Illustration 7, brace A

and brace B are

perpendicular. Find the

values of x and y.

x is 150°

y is 30°

preparing for test 1
Preparing for Test 1
  • For Both Chapters (1 & 2):
    • Read over each Chapter Review (p.69 and p.172)
    • Look through each set of Review Exercises, bypassing the “easy” problems, doing some of the “hard” ones
    • Solving equations graphically won’t be on the test
  • The Test: 3:05pm-4:35pm
    • Be on time to class – no extra time for test if you are late
    • 90 minutes, 20 questions, 100 total points
    • Bring your own #2 pencils & calculator, I supply scratch paper
    • When finished, you may leave the room, but return by 4:45
    • If you miss the test, you must take it in the Math Lab before our next class meeting or you will receive a test grade of 0
    • All tests count toward your grade
test 1 the race is on
Test 1 – the race is on!
  • Next Class:
  • Homework 3.1, 3.2 will be due
  • Test 1 3:05 Sharp(Chapters 1 and 2)
  • Section 3.3 –Solving Systems of 3 Equations