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# Linear Equations

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

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

## 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