Solving Linear Systems in Two Variables

In this lesson, we will do a complete review of how to solve a system of linear equations in two variables using graphing, substitution, and elimination. To solve this type of system using the graphing method, we graph each equation and look for the point of intersection. This point lies on both lines and is, therefore, a solution for the system. Alternatively, we can use an algebraic method to solve such a system. With substitution, we can solve one of the equations for one of the variables. We can then plug in for this variable in one of the original equations. This will give us a linear equation in one variable. Once we solve this equation, we can plug back into one of the original equations and find our other unknown. Additionally, we will look at the elimination method. For this method, we aim to change the equations such that one pair of variable terms are opposites. We want to make sure that each equation is in standard form (ax + by = c) and then we can add the left sides and set this equal to the sum of the right sides. We will find that one variable will drop out and we are left with a linear equation in one variable. From this, we can then solve and substitute back into one of the original equations to find our other unknown. Lastly, we will review special case scenarios such as an inconsistent system or a system with no solution. This occurs when we have two parallel lines. Recall that two parallel lines will never intersect, so there isn't a point that lies on both lines and, therefore, there will never be a solution for this type of system. We will also explore having dependent equations, or a system with an infinite number of solutions. This will occur when a system gives us the same equation twice.