Systems of Equations
When you solve systems of equations you are trying to determine where they
intersect. If both equations are linear they will intersect at one point, be parallel, or
be the same line.
There are four methods to determine the solution, if any, to systems of linear
equations.
You can graph the two equations and see if and where they intersect.
You can use the elimination method. The elimination method as suggested by its
name is where you get one of the variables (X or Y) to cancel when you add the
two equations together. This may require that you alter the coefficients of the
variable you are trying to eliminate. Once you have eliminated one of the variables
you solve for the other variable. Then plug that number into one of the equations to
find the value of the other variable.
You can use the substitution method. The substitution method as suggested by its
name is where you get one of the variables in one of the equations by itself. You
then substitute what that variable is equal to into the other equation. You then solve
the equation for the one variable. Then plug that number into one of the equations
to find the value of the other variable.
You can use matrices and Cramer’s rule.
Example problem worked with all four methods.
2X + Y = 5
X–Y=1
Graph each equation. 2X + Y = 5
-2X
-2X
Y = -2X + 5
Solution ( 2 , 1)
Elimination Method.
2X + Y = 5
X - Y=1
2X + Y = 5
X -Y=1
---------------3X =6
3X/3 = 6/3
X
=2
2(2) + Y = 5
4 +Y=5
4–4+Y=5–4
Y=1
Solution ( 2 , 1)
Substitution Method
2X + Y = 5
X–Y=1
X–Y+Y=1+Y
X=Y+1
2(Y + 1) + Y = 5
2Y + 2 + Y = 5
3Y + 2 = 5
3Y+2–2=5–2
3Y = 3
3Y/3 = 3/3
Y=1
X–Y=1
-X
-X
- Y = -X + 1
-Y/-1 = -X/-1 + 1/-1
Y=X–1
2X + 1 = 5
2X + 1 – 1 = 5 - 1
2X = 4
2X/2 = 4/2
X=2
Solution ( 2 , 1)
Cramer
aX + bY = e
cX + dY = f
d = ad – bc
|a b|
d= |c d|
dx = ed – bf
dy = af – ec
2X+ Y=5
X–Y =1
d = | 2 1 | = 2(-1) – 1(1) = -2 – 1 = -3
| 1 -1 |
dx = | 5 1 | = 5(-1) – 1(1) = -5 – 1 = -6
| 1 -1 |
dy = | 2 5 | = 2(1) – 1(5) = -3
|1 1|
X = -6/-3 = 2
Solution ( 2 , 1)
Y = -3/-3 = 1
|e b|
dx = | f d | dy =
X = dx/d
|a
|c
e|
f|
Y = dy/d
Solve the following systems of equations:
1) X + 2Y = 8
2X + 2Y = 10
2) 3X – Y = 4
2X + Y = 6
3) 2X – 3Y = 1
4X + Y = 9
4) 3X + Y = 8
6X + 2Y = 4
5) 3X + Y = -1
X – 2Y = 9
6) 3X – 2Y = -4
2X + Y = 9
7) 3X – 2Y = 5
X=Y+1
8) 2X – 5Y = 1
Y=X–2
9) 2X + 3 Y = -4
4X + 6Y = -8
10) 3X + 3Y = 3
4X – 2Y = 16
11) X – 2Y = 2
2X + 3Y = 11
12) 2X – 3Y = -3
X + 3Y = 3