Linear Equations

A linear equation is an algebra equation that contains an unknown variable and no exponent greater than 1. In its simplest form, a linear equation is expressed as ax + b = y, where x is the variable and a and b are constants.

Here are two things to keep in mind when you’re solving linear equations:

  1. Isolate the variable in the equation you are trying to solve, which means you work to get it all by itself on one side of the equation.
  2. Whatever operation you perform on one side of the equation, you must do to the other side.

If the equation includes multiple fractions, you can simplify things and save time by eliminating the fractions. Just multiply each fraction by the least common denominator (which is the lowest positive whole number that each fraction’s denominator divides into evenly).

Solutions of Equation

The solutions of an equation with one or more unknowns are those values that make the equation true, or satisfy the equation, when they are substituted for the unknowns of the equation. An equation may have no solution or one or more solutions. If two or more equations are to be solved together, the solutions must satisfy all the equations simultaneously.

Linear Equations

Linear Equations are also called first degree equations. Linear equation involves variables of first degree only (no quadratic or cubic terms).

To solve a linear equation with one unknown, the unknown should be isolated on one side of the equation. This can be done by performing the same mathematical operations on both sides of the equation. For example,

5x - 6 = 12

5x = 18

x = 18/5

Linear equations with two unknowns

Solving for x is simple when it is the only variable, but what if your equation has more than one variable? When you have another equation that contains at least one of the variables, you can solve for either variable. These two equations are called simultaneous equations. You just solve one of the equations for one of the variables and then plug the answer (substitute) into the other equation and solve.

You can also solve simultaneous linear equations by stacking them. This method works when you have as many equations as you have possible variables to solve for. So you can stack these two equations because they contain two variables:

6x + 4y = 66

-2x + 2y = 8

Your goal is to find a way to remove one of the variables.

1. Examine the equations to determine what terms you can eliminate through addition or subtraction.

If you multiply the entire second equation by 3, you can eliminate the x terms in both equations because 2x × 3 = 6x, and 6x - 6x = 0. Just be sure to multiply each term in the equation by the same value. So the second equation becomes -6x + 6y = 24.

2. Combine like terms, and solve for y.

By adding both the equations, you can eliminate the x term and then solve for y. 

10y = 90

y = 9

3. Plug the value of one variable into one of the equations and solve for the other value.

You have found that y = 9, so substitute 9 for the value of y in one of the equations to solve for x.

-2x + 2y = 8

-2x + 2(9) = 8

-2x + 18 = 8

-2x = -10

x = 5

Therefore, the solutions, also referred to as roots, to the simultaneous equations are x = 5 and y = 9.

Number of Solutions

For two linear equations with two unknowns, if the equations are equivalent, then there are infinitely many solutions to the equations. If the equations are not equivalent, then they have either a unique solution or no solution.