Systems of Linear Equations
A system of linear equationsis when more than one equation is con-
sidered at the same time. When two nonparallel, but coplanar, lines are
graphed on the same coordinate graph, they will intersect at exactly one
point. This point of intersection is called the solutionto a system of
equations. At this one coordinate pair, there exists an x- and y-value that
satisfy bothlinear equations. This (x,y) coordinate pair sits on both lines.
If lines are parallel there will not an intersection or a solution to that
system of equations. In this case, when the system is solved algebraically,
the variables will cancel out and you will be left with a false statement, like
4 = –3. This is how you know that they system is unsolvable because the
lines are parallel.
While there are several methods that can be used to find the point that
solves a system of linear equations, only two methods will be presented here.
Method 1: Both of the equations are written in y = mx+ bform.
In this case, set the two equations equal to one another since they both are
equal to “y.” Once you do this, you can combine like terms, solve for x, and
then plug xback into either equation to solve for y.
Example: Find the solution to the system of equations
y= 2x– 20 and y = –4x+ 40
Set the equations equal to each other: 2x– 20 = –4x+ 40
Solve for x: 6x= 60, so x= 10
Now plug in x= 10 to solve for y: y= 2(10) – 20, y= 0
The solution is (10,0), which works in both equations.Method 2: One equation has xor yalone and the other equation
does not.
In this case, use substitution to put the equation that has one isolated vari-
able into the other, more complex equation. Then solve for the existing
variable and plug that answer back into either equation to solve for the
other variable.
Example: Find the solution to the system of equations
x= 3y– 12 and –2y+ 4x= –28
Sub “3y– 12” in for xin the other equation: –2y+ 4(3y– 12) = –28
Solve for y: –2y+ 12y– 48 = –28, so 10y= 20 and y= 2
Now plug in y= 2 to solve for x: x= 3(2) – 12 = 6 – 12, x= –6
The solution is (–6, 2), which works in both equations.501 Geometry Questions