When faced with linear equations for the first time, many people feel overwhelmed and confused by the complexity of mixing numbers and letters to solve the equations. With a few simple guidelines, however, you can learn these fundamental skills used in college algebra and higher mathematics. The methods used to solve onevariable and twovariable linear equations in college algebra are fairly simple.
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OneVariable Linear Equations

Recall inverse relationships, such as 1 and 1, and 1/3 and 3, to solve onevariable linear equations. The solutions require using the inverse relationships of addition and subtraction, and multiplication and division.

Isolate the variable "x" on one side of the equation. If x = y, then x + a = y + a. Based on this logic, use inverses to move values from one side of an equation to the other side of the equation.
To isolate x by using the inverse of subtraction in the equation x  5 = 8, add the inverse of 5, which is +5, to both sides of the equation. The result is: x  5 + 5 = 8 + 5. The solution is: x = 13.
To use the inverse of addition in the equation x + 9 = 12 to isolate x, subtract the inverse of +9 from both sides of the equation. The resulting equation is: x + 9  9 = 12  9. After subtracting 9 from both sides of the equation, you will find that x = 3.
Using the inverse of division in the equation (1/2)x = 10 to isolate x requires multiplying the inverse of 1/2 by both sides of the equation. The resulting equation is: (1/2)(2) = 10(2). Multiplying both sides of the equation by 2 reveals that x = 20.
To isolate x by using the inverse of multiplication in the equation 4x = 8, divide both sides of the equation by 4. The resulting equation is: 4x/4 = 8/4. The solution is: x = 2.

Check the solution. Plug the solution into the original equation to verify that its value is correct. If the original equation is x  5 = 8 and you found that the value of x is 13, for instance, then check the solution by simply using the value 13 instead of x in the original equation. The equation then becomes 13  5 = 8 or 8 = 8, which is the correct answer.
TwoVariable Linear Equations  Addition/Elimination Method

Choose a variable to eliminate in a twovariable linear equation such as 4x  10y = 32 and 6x + 4y = 10. To eliminate "x," multiply the equations by common multiples to obtain equal but opposite values of x: 3(4x  10y = 32) and 2(6x + 4y = 10). The example will then look like this: 12x  30y = 96 and 12x  8y = 20.

Add the equations together to eliminate x. An example is:
12x  30y = 96
12x  8y = 20
38y = 76

Solve for y in the equation 38y = 76. The process is:
38y/38 = 76/38
y = 2
y/1 = 2/1
y = 2

Plug the value of y into the original equations, and find the value for x. The first original equation is 4x  10y = 32, and the solution process is:
4x  10(2) = 32
4x + 20 = 32
4x + 20  20 = 32  20
4x = 12
4x/4 = 12/4
x = 3
The second original equation is 6x + 4y = 10. Its solution process is:
6x + 4(2) = 10
6x  8 = 10
6x  8 + 8 = 10 + 8
6x = 18
6x/6 = 18/6
x = 3

Check the solutions y = 2 and x = 3 for the original equations, 4x  10y = 32 and 6x + 4y = 10. The process for the first equation is:
4(3)  10(2) = 32
12 +20 = 32
32 = 32
The process for the second equation is:
6(3) + 4(2) = 10
18  8 = 10
10 = 10
Twovariable linear equations can have one solution, no solution or many solutions. That is why it is very important to check solutions in the original equations.
References
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