A **Linear System** is a set of 2 or more **Linear Equations**. It is recommended you first view this lesson to better understand how linear equations work.

In a linear system, there are 3 unique cases for intersection:

- Lines don't intersect; this means that they are parallel to one another:
- Lines only intersect at one point; this is the most common form of linear system:
- Lines intersect infinetely at every point; this means that the lines/equations are the exact same:

There are 3 main methods for solving a linear system:

- 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

Graph the lines \(y = 5x - 7\) and \(y = -3x + 1\) to find the Point of Intersection.

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

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

\(y = 8x - 10\)

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

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

Solve this and you have the \(x\)-coordinate of the Point of Intersection:

\(-5x = -1\)

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

\(x = 0.2\)

Plug \(x\) (\(0.2\)) into either equation to determine the corresponding \(y\)-coordinate and the Point of Intersection:

\(1.6 - y = 10\)

\(- y = 10 - 1.6\)

\(-y = 8.4\)

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

\(y = -8.4\)

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

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

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

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

\(y = -x + 8\)

Add or subtract a multiple of one equation to (or from) the other equation so that the terms cancel each other out:

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

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

\(4y = 20\)

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

\(y = 5\)

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

\(-3x = -9\)

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

\(x = 3\)

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

Find the Point of Intersection between the linear equations \(y = 7x + 3\) and \(x - y = 3\) using any algebraic method.

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