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What is a System of Linear Equations?

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Frustrated Student: “This is why I hate Algebra. I'm not going to use this; I’m going to become an actor.”

Patient Tutor: "You're right. Unless you're Russell Crowe in A Beautiful Mind or Matt Damon in Good Will Hunting, you can leave quadratic functions in the classroom. Patience, discipline, decision making skills - these are the lessons you learn from  quadratic functions that you use in real life."

1. Systems of Equations Definitions

  • System of linear quations:  Two or more equations with two or more variables. When graphed, these lines intersect at some point (x, y). Note: This lesson focuses on systems of equations with 2 equations and 2 distinct variables.

  • Intersection: The point (x,y) where the lines meet.

  • Solution: The x and y value of the intersection of the 2 lines.

  • No Solution: When graphed, sometimes the system of equations is actually a set of parallel lines. Because parallel lines never intersect, this system will have no solution.

  • Infinite Number of Solutions: When graphed, this system of equations is actually the same line.

2. Systems of Linear Equations Differ from Regular Equations

Regular Solving

Purpose: Find the value of 1 variable.

Example

A puppy weighs 16 pounds. When a kitten sits on the scale with the puppy, the combined weight of the animals is 26 pounds. How much does the kitten weigh?

Equation

16 + k = 26, where k is the weight of the kitten in pounds.

Solving a System of Linear Equations

Purpose: Find the value of 2 variables.

Example

The combined weight of a puppy and a kitten is 180 pounds.  The sum of 3 times the weight of the puppy and twice the weight of the kitten is 414 pounds. Find the weight of each animal.

Equations

p + k = 180  (The puppy’s weight, p, plus the kitten’s weight, k.)

3p + 2k = 414 (The sum of 3 times the weight of the puppy and twice the weight of the kitten.)

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