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Unit 5: Systems of Equations

Lesson 1: Systems of Linear Equations

Systems of Equations: 2 or more linear equations that use the same variables

Solution to a System of Equations: any point (x, y) that makes both equations

true

Solve the system by testing an ordered pair:

Example: Is the point (1, -4) a solution to the system of equations?

x + y = -3

2x – y = 6

Substitute 1 for x and -4 for y in each equation:

1 + (-4) = -3 2(1) – (-4) = 6

-3 = -3 2 + 4 = 6

6 = 6

Systems of Equations Graphing Example: the solution is where the lines intersect:

(-1, 1) is the only point that is on BOTH lines

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Notes and Examples from Class:

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Lesson 2: Solving Systems Using Inspection

Systems of Equations have 1 Solution when the lines intersect in 1 place

Example:

Systems of Equations have 0 Solutions when the lines never intersect

Example:

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Systems of Equations have Infinite Solutions when the lines are entirely on top of

each other

Example:

Classification of Systems of Equations:

Consistent: if the system has at least 1 solution

Independent: if the system has exactly 1 solution

Dependent (or Coincident) : if the system has an infinite number of solutions

Inconsistent: if the system has no solutions

You can often classify systems of equations just by looking at them:

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Notes and Examples from Class:

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Lesson 3: Using Graphs to Solve Systems Step 1: Write the equations in Slope-Intercept form

Step 2: Graph each equation using the slope and y-intercept

Step 3: Identify the point of intersection

Step 4: Test this point in BOTH equations to verify the solution

Example: Rewrite the first equation in Slope Intercept Form:

Equation 1: x + y = 5

y = -x + 5 or y = -1x + 5 Slope = -1, y-Intercept = 5

Equation 2: y = ½ x – 1 Slope = ½, y-intercept = -1

If the point of intersection is not exactly on the line, you may be asked to estimate

the solution.

Example: the solution is approximately (-1, 2)

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Notes and Examples from Class:

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Lesson 5: Substitution Method

Substitution uses Algebra to solve systems of equations instead of a graph.

Follow these steps:

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Special Cases

Notes and Examples from Class:

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Lesson 8: Core Focus: Applications of Linear Systems

Example of a Real Life Problem using Systems of Equations:

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Notes and Examples from Class:

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Lesson 9: Applications of Linear Systems

Application Example:

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