Linear Systems

A Linear System is a set of 2 or more Linear Equations that are analyzed at the same time. It is recommended you first view this lesson to better understand how linear equations work.

The point at which 2 lines cross is called the Point of Intersection. In a linear system, there are 3 unique cases for intersection:

1. Lines don't intersect; this means that they are parallel to one another:


2. Lines only intersect at one point; this is the most common form of linear system:


3. Lines intersect infinetely at every point; this means that the lines/equations are the exact same:

Methods for Solving Linear Systems

There are \(3\) main methods for solving linear systems:

Graphing Method

• The simplest but least accurate way of portraying the relationships between the set of lines.
• All you need to do is graph the set of lines based on their resepctive equations to see if or where they intersect
• Before graphing the lines, we can create a table of values to identify where each point of the respective lines lie

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Substitution Method

Example

Using the substitution method, we can solve the linear system with linear equations \(8x - y = 10\) and \(3x - y = 9\)

First, we can solve one linear equation for \(x\) in terms of \(y\):

\(8x - y = 10\)

\(y = 8x - 10\)

Next, we can substitute that expression for \(y\) in the other linear equation:

\(3x - (8x - 10) = 9\)

\(3x - 8x + 10 = 9\)

Then, we can solve this system to determine the \(x\)-coordinate of the Point of Intersection:

\(3x - 8x = 9 - 10\)

\(-5x = -1\)

\(\cfrac{\cancel{-5}x}{\cancel{-5}} = \cfrac{-1}{-5}\)

\(x = 0.2\)

Finally, we can plug P\(x\), \(0.2\), into either equation to determine the corresponding \(y\)-coordinate and the Point of Intersection:

\(8(0.2) - y = 10\)

\(1.6 - y = 10\)

\(- y = 10 - 1.6\)

\(-y = 8.4\)

\(\cfrac{\cancel{-}y}{\cancel{-1}} = \cfrac{8.4}{-1}\)

\(y = -8.4\)

Therefore, we can determine that the Point of Intersection is \(\boldsymbol{(0.2, -8.4)}\).

NOTE: You can also start by solving for \(x\) in terms of \(y\) using the same steps outlined above.

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Elimination Method

Example

Using the elimination method, we can solve the linear system with linear equations \(4 + y -3x = 0\) and \(x + y = 8\)

First, we can align all the terms in both equations so they share the same form (ie slope-intercept form):

\(y_1 = 3x - 4\)

\(y_2 = -x + 8\)

Next, we can add or subtract a multiple of one equation to (or from) the other equation so that the terms cancel each other out:

\(3(y = -x + 8)\)

\(3y = -3x + 24\)

\((3y + y) = (-3x + 3x) + (24 - 4)\)

\(4y = 20\)

\(\cfrac{\cancel{4}y}{\cancel{4}} = \cfrac{20}{4}\)

\(y = 5\)

Then, we can solve for \(x\) by substituting the \(y\)-value, \(5\), into one of the original equations:

\(4 + 5 -3x = 0\)

\(-3x = -9\)

\(\cfrac{\cancel{-3}x}{\cancel{-3}} = \cfrac{-9}{-3}\)

\(x = 3\)

Therefore, we can determine that the Point of Intersection is \(\boldsymbol{(3, 5)}\).

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