A system of linear equations consists of multiple equations that use the same variable set, and which represent lines that exist on the same plane. A solution to a system of linear equations is the number or numbers that solve all of the equations at the same time, and represents the point at which the lines would cross if graphed. There are three ways to solve a system of equations that do not require the use of a calculator.
Isolation and Substitution
When trying to solve a system of equations, begin by trying to isolate one of the variables. For example, if you are trying to solve the system y + 6x = 12 and 5x = 15 - 5y, the easiest variable to isolate is y in the first equation. To isolate y, move everything else in the problem to the opposite side of the equal sign. In the example, subtract 6x from both sides to get the rewritten equation y = 12 - 6x. Substitute this for y in the second equation. This will give you a new equation: 5x = 15 - 5(12 - 6x). Solve the new equation by isolating x; distribute the number outside the parentheses first. In the example, distributing the five would result in the equation 5x = 15 - 60 + 30x. Simplify the equation as much as possible. You can combine the 15 and -60 to get -45. Move the variables to one side of the equation and the numbers to the other side by performing opposite operations. In the example we would the resulting equation would be 45 = 25x. Solve for x by dividing by the coefficient in front of x. The example results in 1.8 = x.
To finish solving the equation, substitute the numerical solution for "x" back into one of the original equations and solve. In the example, 1.8 is substituted into y + 6x = 12 to get y + 6(1.8) = 12. This simplifies to y + 10.8 = 12. Getting y alone yields y = 1.2 So the solution for the system of equations is (1.8, 1.2).
Addition or Subtraction
Solve the system of equations by adding the two systems together to eliminate one variable. First, multiply one of the equations by the common element that will allow you to eliminate one variable. For example, if you are solving the system of equations 4x - 4y = -4 and 3x + 2y = 12, both of the y variables are factors of 2. Multiplying the second equation by two will result in the equation 6x + 4y = 24, and allow you to cancel out the y terms. Add or subtract the two equations from each other. In the example, the matching elements have different signs, so you add the equations. If they were both positive or both negative you would subtract. The resulting equation is 10x = 20. Divide both sides by the coefficient to isolate the variable. In the example, this yields x = 2. Substitute this value back into one of the original equations and solve for the other variable. In the example, this yields 4(2) - 4y = -4. This simplifies to 8 - 4y = -4. Move the eight to the opposite side of the equation to get -4y = -12. Divide by the coefficient. For the example, this results in a solution of y = 3. The solution to this system of equations is written as (2, 3).
Graph the equations using the x and y axes. Set up each equation in slope-intercept form. Slope-intercept form is y = mx + b, where "y" represents the slope of the line (vertical rise over the horizontal run) and "b" represents the y-intercept for the line. An equation can be put on this form by isolating "y" on one side of the equation. For example, the system x + y = 3 and x - y = 1 would be rewritten y = -x + 3 and y = x - 1. Graph the lines on the x and y axes. The solution for the system of equations is the point where the two lines intersect.
Not all systems of equations have one solution. If the lines are parallel, there is no solution. When graphing, these lines have the same slope. When solving by substitution or addition and subtraction, both variables will be eliminated, and an obviously wrong solution will remain. For example, you might get the answer 5 = 7. If the equations represent the same line, written in two different forms, the number of solutions will be infinite. Graphed, the lines would overlap completely. When working with substitution or addition and subtraction, you would eliminate everything, resulting in the answer 0 = 0, 5 = 5 or any other number equal to itself.