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Solving an Algebraic Linear Equation with One Variable
by Ron Kurtus (revised 17 August 2012)
A linear equation with one variable consists of numbers or constants and multiplies of a variable. The standard form of such an equation is ax + b = 0, where a and b are constants and x is the variable. Often, the equation is in a more complex form. The solution of the equation is found by operating on both sides of the equation to get it into the form similar to x = −b/a.
Questions you may have include:
- How do you operate on the equation?
- How do you solve for x?
- What happens if the equation is in a more complex form?
This lesson will answer those questions.
Rules for solution
When you have a linear equation with one variable, your goal is to manipulate the expressions, such that you you end up with the variable x on the left side of the equal sign and the constants on the right side. That is solving the equation.
For example, the solution of the equation 4a = 3 − x is x = 3 − 4a.
Basic rule
The basic rule used in solving equations in Algebra is:
What you do on the left side of the equal sign, you must do on the right side.
If you add a term on the left side, you must add the same term on the right side. If you multiply a term on the left side, you must multiply the same term on the right side.
Examples
In the equation 4a = 3 − x, you want to get the x on the left side and the other items on the right side. You perform the following operations:
Add x to both sides of the equation.
4a + x = 3 − x + x
4a + x = 3
Subtract 4a from both sides of the equation.
4a − 4a + x = 3 − 4a
x = 3 − 4a, which is the solution to the equation.
Solving by combining like terms
You can solve an equation like 2x + 3 = −4x − 7 first getting all the x terms on the left side and all the constant terms on the right side. Next, you combine like terms. Then you divide by the multiple of x to get your solution.
Example
Consider the equation:
2x + 3 = −4x − 7
Add 4x to both sides.
2x + 4x + 3 = −4x + 4x − 7
Combine like terms.
6x + 3 = −7
Subtract 3 from both sides.
6x + 3 − 3 = −7 − 3
Combine like terms.
6x = −10
Divide both sides by 6.
6x/6 = −10/6
Simplify the fraction.
x = −5/3 or x = −1 2/3
Note: It is a good idea to go step-by-step instead of trying to do several things at once or to do things in your head.
Another example
Consider the equation:
2x/3 + 3 − x = 2(x + 2) − 5
Multiply out to get rid of the parentheses.
2x/3 + 3 − x = 2x + 4 − 5
2x/3 + 3 − x = 2x − 1
Get rid of the fraction by multiplying both sides by 3.
3(2x/3 + 3 − x) = 3(2x − 1)
Multiply out to get rid of the parentheses.
2x + 9 − 3x = 6x − 3
Combine like terms.
9 − x = 6x − 3
Subtract 9 from both sides.
−x = 6x − 12
Subtract 6x from both sides.
−7x = −12
Divide by −7.
x = 12/7 or x = 1 5/7
Variable in a fraction
There are equations where the x term is part of the denominator in an equation. In such a case, you must multiply both sides of the equation by the x term, so that it does not contain variable fractions. Likewise, you want to remove any fractions in the equation but multiplying by the denominator of the equation.
Example
Consider the equation:
2x/(x + 1) = 7/12
Multiply both sides by (x + 1).
2x(x + 1)/(x + 1) = 7(x + 1)/12
Simplify the fraction (x + 1)/(x + 1) = 1.
2x = 7(x + 1)/12
Multiply both sides by 12.
24x = 7(x + 1)
Multiply with distributive law or multiply out to get rid of the parentheses.
24x = 7x + 7
Subtract 7x from both sides.
24x − 7x = 7x − 7x + 7
Combine like terms.
17x = 7
Divide by 17 to get solution of equation.
x = 7/17
Another example
Consider the equation:
1/(5x − 3) = 3/x
Multiply both sides by (5x − 3).
1 = 3(5x − 3)/x
Multiply both sides by x.
x = 3(5x − 3)
Note that sometime these two steps are combined and called "cross multiplying" the equation. One problem is that shortcutting can result in mistakes. Also, it is better to know what you are doing and why for better understanding.
Multiply with distributive law (remove parentheses).
x = 15x − 9
Subtract 15x from both sides.
−14x = −9
Divide both sides by −14x.
x = 9/14
Summary
A linear equation with one variable consists of numbers or constants and multiplies of a variable. The standard form of such an equation is ax + b = 0, where a and b are constants and x is a variable. Often, the equation is in a more complex form.
The solution of the equation is found by operating on the equation to get it into the form similar to x = −b/a. In other words, you want the x alone on the left side and the other items on the right side of the equation. The rule is what you do on the left side, you do on the right side.
Go step-by-step
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Solving an Algebraic Linear Equation with One Variable