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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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