# Linear Equations

**Topics:**Elementary algebra, Derivative

**Pages:**6 (550 words)

**Published:**November 11, 2014

Linear Equations

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Linear Equations

A linear equation is an equation for a straight line

These are all linear equations:

y = 2x+1

5x = 6+3y

y/2 = 3 x

Let us look more closely at one example:

Example: y = 2x+1 is a linear equation:

The graph of y = 2x+1 is a straight line

When x increases, y increases twice as fast, hence 2x When x is 0, y is already 1. Hence +1 is also needed So: y = 2x + 1

Here are some example values:

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x

y = 2x + 1

1

y = 2 × (1) + 1 = 1

0

y = 2 × 0 + 1 = 1

1

y = 2 × 1 + 1 = 3

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Linear Equations

2

y = 2 × 2 + 1 = 5

Check for yourself that those points are part of the line above!

Different Forms

There are many ways of writing linear equations, but they usually have constants (like "2" or "c") and must have simple variables (like "x" or "y").

Examples: These are linear equations:

y = 3x 6

y 2 = 3(x + 1)

y + 2x 2 = 0

5x = 6

y/2 = 3

But the variables (like "x" or "y") in Linear Equations do NOT have: Exponents (like the 2 in x2)

Square roots, cube roots, etc

Examples: These are NOT linear equations:

y2 2 = 0

3√x y = 6

x3/2 = 16

Slope-Intercept Form

The most common form is the slopeintercept equation of a straight line :

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Linear Equations

Slope (or Gradient)

Y Intercept

Example: y = 2x + 1

(Our example from the top, which is in SlopeIntercept form) Slope: m = 2

Intercept: b = 1

Play With It !

You can see the effect of different values of m and b at Explore the Straight Line Graph

Point-Slope Form

Another common one is the PointSlope Form of the equation of a straight line:

y y1 = m(x x1)

Example: y 3 = ¼(x 2)

x1 = 2

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Linear Equations

y1 = 3

m = ¼

General Form

And there is also the General Form of the equation of a straight line:

Ax + By + C = 0

(A and B cannot both be 0)

Example: 3x + 2y 4 = 0

A = 3

B = 2

C = 4

There are other, less common forms as well.

As a Function

Sometimes a linear equation is written as a function , with f(x) instead of y:

y = 2x 3

f(x) = 2x 3

These are the same!

And functions are not always written using f(x):

y = 2x 3

w(u) = 2u 3

h(z) = 2z 3

These are also the same!

The Identity Function

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Linear Equations

There is a special linear function called the "Identity Function":

f(x) = x

And here is its graph:

It makes a 45° (its slope is 1)

It is called "Identity" because what comes out is identical to what goes in: In

Out

0

0

5

5

2

2

...etc

...etc

Constant Functions

Another special type of linear function is the Constant Function ... it is a horizontal line:

f(x) = C

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Linear Equations

No matter what value of "x", f(x) is always equal to some constant value.

Using Linear Equations

You may like to read some of the things you can do with lines:

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Finding the Midpoint of a Line Segment

Finding Parallel and Perpendicular Lines

Finding the Equation of a Line from 2 Points

Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10

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