Test 3 Review (Ch. 4) Math 102
Ch. 4: Systems of linear equations in two variables.
4.1: Introduction to linear systems/graphical solutions.
1. You should know what a linear system of equations means: we are looking for the unique
ordered pair that solves both equations.
2. Since each equation in a linear system represents a line, this means that we want the unique
point sitting on both lines. Thus, we find a graphical solution to a system by graphing each
equation in the x-y plane and looking for the intersection.
3. Special cases: if two lines are parallel, then there is no intersection (so there is no solution to
the system). If the two equations in the system are equivalent (so that each one actually
represents the same line), then we say “the lines coincide” and there are an infinite number of
solutions. You can also say “any point on the line”.
Note: you should be able to quickly graph both lines in the system by solving for y and using
slope-intercept form.
4.2: Elimination method for solving linear systems.
1. Elimination is based on the fact that we can “add equations”, meaning that we can add the left
sides of two equations and the right sides of the same equations to generate a new equation.
2. If we can eliminate a variable by adding two equations, we are left with a new equation that
has only one variable. This allows us to solve for that variable (either x or y). Once we have the
x or y coordinate of the solution point, we can get the other coordinate by substituting that value
into either of the original equations.
3. We usually have to “rig” one or both equations in order for a variable to be eliminated when
we add the equations. We do this by multiplying both sides of one (or both) equations by the
proper constant(s) so that x or y is eliminated when we add the equations.
4. Note that systems involving fractions or decimals can be simplified by multiplying both sides
of the offending equation by the least common denominator.
5. If an answer like 0=2 results from elimination, it means there is no solution to the system (the
lines are parallel). If an answer like 1=1 results, the lines coincide and there are an infinite
number of solutions.
