What is a Linear Equation?
A linear equation is an expression that can be written in the form ax+b=0, where a and b are constants and x represents an unknown variable that, once found, will make the expression true.
The following are some basic examples of linear equations having one variable:
Linear Equation Examples
7x + 3 = 202(y + 1) - 4 = 4(y - 3)
16.5a + .25a = 25.1a
| 1 | x | + | 3 | = | 2 | x | - | 4 | ||
| 2 | 4 | 5 |
How to Solve Linear Equations
Solving linear equations is a process aimed at getting the variable by itself on one side of an equal sign and everything else on the other side. To explain how to solve linear equations, I will use an example equation that contains all 4 types of terms that can be handled by the linear equation solver.
| Example equation: | ||||||||||||||||||
| 3(x-1)+1/2=.25+4 | ||||||||||||||||||
| First use the distributive property to get rid of the parenthesis: | ||||||||||||||||||
| 3(x-1) = 3x-3 | ||||||||||||||||||
| After removing parenthesis we are left with: | ||||||||||||||||||
| 3x-3+1/2=.25+4 | ||||||||||||||||||
| Remove the fractions by multiplying both sides of the equation by 2 (least common multiple of all denominators in equation): | ||||||||||||||||||
| 2(3x-3+1/2)=2(.25+4) | ||||||||||||||||||
| After multiplying both sides by 2 we are left with: | ||||||||||||||||||
| 6x-6+1=0.5+8 | ||||||||||||||||||
| Remove decimals by multiplying both sides of the equation by 100: | ||||||||||||||||||
| 100(6x-6+1)=100(0.5+8) | ||||||||||||||||||
| After multiplying both sides by 100 we are left with: | ||||||||||||||||||
| 600x-600+100=50+800 | ||||||||||||||||||
| Combine the like terms: | ||||||||||||||||||
| -600 + 100 = -500 and 50 + 800 = 850 | ||||||||||||||||||
| After combining like terms we are left with: | ||||||||||||||||||
| 600x-500=+850 | ||||||||||||||||||
| Add 500 to both sides of the above equation to get constants on right side: | ||||||||||||||||||
| -500 + 500 = 0 and 850 + 500 = 1350 | ||||||||||||||||||
| After adding 500 to both sides we are left with: | ||||||||||||||||||
| 600x=1350 | ||||||||||||||||||
| Get the variable by itself by dividing both sides by 600: | ||||||||||||||||||
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| After dividing both sides by 600 we are left with: | ||||||||||||||||||
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| Simplify the result: | ||||||||||||||||||
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Always Check to Make Sure Your Answer is a Solution
Regardless of which method or tool you use to solve linear equations, you should always check your results by substituting the result for the variable and seeing if the resulting equation proves to be true. After all, humans create math solvers, and humans make mistakes.
To make sure our answer to the above example linear equation is, in fact, a solution, let's substitute 9/4 for x and check to see if the resulting equation proves to be true:
Substitute Answer for Variable to Check Your Work
3(9/4 - 1) + 1/2 = .25 + 44(27/4 - 3/1 + 1/2) = 4(4.25)
27 - 12 + 2 = 17
15 + 2 = 17
17 = 17
The solution 9/4 is the solution to the example equation.
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