Simultaneous equations can be solved by two types of algebraic manipulation: substitution and addition. Either method will work for any problem but one method is usually faster for a given problem. This allows advanced students to tailor their solutions to the information in the problem. In substitution, one variable is expressed in terms of the other and that expression is then substituted for the variable in the second equation. With addition, the two equations are added together so as to cancel out one variable.
For students who prefer visual solutions as opposed to algebraic manipulation, simultaneous equations can be solved by graphing each equation separately. The point of intersection of the two lines is the only pair of (x, y) values that satisfies both equations. For example, the lines y = 7 - x and y = 4 + x intersect at (1.5, 5.5). Thus, x = 1.5 and y = 5.5.
The rule that solving for some number of variables requires that number of equations only holds true for unique equations, i.e., those that are not merely rearrangements of each other. Thus, students must check that the two equations are not equivalent to each other. For example, x - y = 5 is equivalent to 2y - 2x = -10, as multiplying the first equation by -2 yields the second equation.
Because simultaneous equations cannot be solved individually, they require more steps and planning to solve than do single-variable equations (equations with one variable each). As stated, students must first check that the simultaneous equations are not equivalent to each other. When using addition, students must also multiply one equation by the correct number in order to cancel out one variable.