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LinearEquations SolvingInequalities MultiplyingPolynomials item2a
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Linear Equations

Thanks to René Descartes, the French philosopher and mathematician, who supposedly was lying in bed and staring up at a fly on his ceiling, we have a fabulous way to graph equations. We now have what is known as the Cartesian Plane! We can use an ordered pair of coordinates to specify a point to indicate a specific orientation or location in space.

Lesson 1: 'Basic' Linear Equation

Before you can do anything, you need to know the basics. Understanding the basics is key as this will be your foundation.

Number Line: A number line has arrows at both ends which means it goes on indefinitely.


x-axis and y-axis: two number lines, one goes horizontally (x-axis) and the other vertically (y-axis).

The x- and y-axes give you dimensionality, so points can be plotted in a plane.

When plotting points, you move across the x-axis first then, you move up or down along the y-axis.

Think: x comes before y in the alphabet to help you remember this.

coordinate pair (or ordered pair): how to describe a point's location on the Cartesian grid by the point of intersection on the x-axis and y-axis; (x, y)

abscissa: the value of the x in the ordered pair (x, y)

ordinate: the value of the y in the ordered pair (x, y)

Point A is located at (4, 4)
Point B is located at (–3, –5)

function: a table of ordered pairs that shows the relation between the domain (x values) and the range (y values) and how they correspond to one another.




Hint: An easy way to help you with the ordered pairs is to put parenthesis around the numbers in the columns. Check it out…
Now, the function table has given the ordered pairs that can be graphed!
(0, 4), (1, 5), and (-2, 2)

graphing the function: plotting the points of the function on a Cartesian plane/grid

All the points on the 'graph' are the solutions to the equation. This is the awesome part of graphing linear functions.

Look at a spot where the blue line crosses the intersection of two axes…(-5, -1) is one such place. Now, if you go back to the equation and use the Substitution Property, replacing the x with -5, guess what! y is -1! Isn't that the coolest thing?
Lesson 2: Linear Equation 'Forms'

There are three different 'forms' of linear equations. You should be familiar with all three because you will be working with problems in all of the forms.

Slope-Intercept Form: This is the form you learned in pre-algebra and probably the one you think of when you hear the term 'linear equation.'

b is the y-intercept

y = mx + b Example: y = 6x +

m is the slope

Standard Form: This form allows us to write equations for vertical lines (which have an undefined slope), is useful when solving systems of linear equations, and simplifies finding parallel and perpendicular lines.
Example: 7x + 3y =
Ax + By  =

A and B are both not zero

Point-Slope Form: This form is useful when you need to find the equation of a line and you know two points. You can determine the slope from the two points then simply substitute one of the ordered pairs into the equation.

m is the slope

(y – y1)  = Example: (y – 7) =

(x1, y1) is a point on the graph of the equation

Lesson 3: write an Equation

You can always use a function table to write an equation, too! This relies on patterning (yep, all that pattern work from Kindergarten comes in handy now…).

Here we have a table and you have to come up with the equation (function rule) that fits.

Step 1: Find the relationship between each x and each y value.

Begin by finding the patterns of change.

There is a difference of 3 between each y value.

There is a difference of 2 between each x value.


Step 2: Determine the rate of change.

Write the pattern you find as a fraction.
This is now your slope!

Step 3a: Choose any coordinate point from the table. Use the x and y values from your chosen point to write an equation using the point-slope form. (y y1) = m (x x1)


Step 3b: IF you happen to have a point where x = 0, look at your y value. This will be your y-intercept. You can now use the slope-intercept form instead. y=mx + b

That's it!

From our example…

(y – 6)  =
using the point-slope form (y
using the slope-intercept form y

©2011–2017 Sherry Skipper Spurgeon.

All Rights Reserved.

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