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Linear Equations

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Linear Equations are mathematical expressions of anything that proceeds without varying, shifting or altering direction along its track or path.

Graphing Linear EquationsLinear Equations In Two VariablesSystem of EquationsLinear Function

They are first-degree polynomials and graph as straight lines. Their variables do not appear in the denominator of a fraction and have an exponent of 1:

4x - 12 = 16 is an example.

If a linear equation solution is two variables it provides Cartesian Coordinates, (x, y), to graph a straight line that is sloping, horizontal or vertical. Multiple lines on a Cartesian coordinate system diagram can be parallel or intersecting. A straight line requires at least 2 coordinates, each specifies a unique point on that line.

Sometimes “Rectangular” is used instead of Cartesian as; Rectangular Coordinate System, Rectangular Coordinates.

Single Variable Linear Equations

Are solved by isolating one variable onto one side of the equation:

2(3y - 1) = 4(y - 5)

6y - 2 = 4y - 20

6y - 4y = 2 - 20

2y = -18

y = -9

2x/4 = 3(x - 4)/2 =

4 (2x / 4) = 4 [3 (x - 4) / 2]

2x = 2 [3 (x - 4)]

2x = 6x - 24

x = (6x - 24) / 2

x = 3x - 12

x - 3x = -12

-2x = -12

x = 6

0.4 x + 0.37 = 0.21x - 4

100 (0.4 x + 0.37) = 100 (0.21x - 4)

40x + 37 = 21x - 400

40x - 21x = -400 - 37

19x = -437

x = -23

Literal Equations

If an equation has more than one variable; solve for any one variable by isolating that variable onto one side of the equation. Many formulas are Literal Equations.

4c + 2x = 5b

2x = 5b - 4c

x = (5b - 4c)/2

2 (2x - b) = 2c/3

6 (2x - b) = 2c

12x - 6b = 2c

x = (6b + 2c)/12

x = 2 (3b + c)/12

x = (3b + c)/6

Linear Equation Word Problems

Creating equations to solve linear problems.

A formula for velocity is: V = K + 32t; where V is velocity, K is energy and t is time:

V - K = 32t

(V - K)/32 = t

Cut a pipe that is 21 feet long so that one piece is twice the length of the other:

p + 2p = 21

3p = 21

p = 7

The shorter pipe is 7 feet. The longer piece is 14 feet.

7 + 14 = 21

The sum of two numbers is 17. Find the numbers if twice the smaller is 4 more than the larger:

x + 2x - 4 = 17

3x = 21

x = 7 is the smaller number.

2x - 4 = 14 - 4 = 10 is the larger number.

Help: First break the problem into components then solve it;

State the equation as f(g) + f(h) = 17 then define f(g) and f(h).

f(g) = x, the smaller number.

f(h) = 2x - 4, second number is four less twice the smaller number.

f(g) + f(h) = 17 = x + 2x - 4

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