Systems of Linear Equations in Two Variables

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Section 4.1

• Systems of Linear Equations in Two Variables

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Introduction

• In this section we will explore systems of linear
  equations and their solutions.
• A system of linear equations is in the form

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Solutions

• We know from Chapter 4 that the graph of every
  linear equation is a straight line.
• When you have two linear equations, you have (at
  most) two lines.
• There are three possible scenarios for the
  relationship between those lines

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1. The lines intersect in a single point

• The system will have one ordered pair solution.

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2. The two lines are parallel.

• There are no ordered pair solutions.

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3. The two lines are actually the same line.

• There are infinitely many ordered pair solutions.

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Solving Methods

• Substitution
• One of the equations has an isolated variable, or
  a variable that can be easily isolated.
• Substitute what the variable is equal to into the
  other equation. Solve the resulting equation.
• Use that solution to find the other variable.

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Examples
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Solving Methods

• Elimination
• Put both equations into standard form.
• If necessary, multiply one or both equations by
  some number(s) to create a set of opposite
  coefficients.
• Add the equations together. One variable will
  cancel. Solve for the remaining variable.
• Substitute into either equation to find the other
  variable.

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Examples
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Special Cases

• Both variables cancel out.
• If the resulting statement is true, you have
  infinitely many solutions (the two equations make
  the same line).
• If the resulting statement is false, you have no
  solution (the two equations make parallel lines).

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Examples