Systems of algebraic equations with two variables may contain either one, infinitely many or no solutions. A system will have one solution if the graphs of the equations are intersecting lines. A system will have infinitely many solutions if the equations have the same graph. A system will have no solution if the graphs of the equations are parallel lines. Solve a system of equations using substitution to isolate and solve for the value of its variables.
Isolate one of the variables in one of the equations by adding, subtracting, multiplying and dividing values from the same side of the equation to cancel out all terms except for the variable. For example, in the system of equations 2x + 4 = 2y, x + 6 = 2y, isolate the y in the second equation by subtracting 6 from both sides of the equation to get x + 6 - 6 = 2y - 6, which simplifies to x = 2y - 6.
Substitute the expression on the right side of the equation from Step 1 in for its corresponding variable in the other equation. In the above example, substitute the expression "2y - 6" in for the value of x in the equation 2x + 4 = 2y, yielding the equation 2(2y - 6) + 4 = 2y, or 4y - 12 + 4 = 2y.
Solve the equation from Step 2 for the single variable by combining like terms and isolating the variable using the same methods as in Step 1. In the above example, combine like terms to get the equation -8 = -2y. Divide both sides of the equation to isolate the variable: y = 4.
Substitute the value for the variable you solved in Step 3 into the other equation and solve for the second unknown variable. In the above example, plug the value y = 4 into the equation x + 6 = 2y to get x + 6 = 8. Subtract 6 from both sides of the equation to get x = 2. The solution to the system of two equations is x = 2, y = 4.