Transcript Chapter 3: Systems of Equations

Chapter 3: Systems of Equations
A System of Equations is two or more equations using the same variables.
- The solution to a system is the ordered pair (x, y) that satisfies BOTH of
the equations.
-
There are 3 methods used to solve a system of equations
1. Graphing
2. Substitution
3. Elimination
Solve by Graphing
- Graph both lines to find the point of intersection.
1. One solution
(x, y)
2. Parallel Lines
have no solution
3. If both equations are
graphed as the same
line, there are infinitely
many solutions
Solve by Substitution
-One of the equations in the system must be in the form x = something or
y = something in order to substitute.
EXAMPLE: Solve the system
Substitute 2y in for x in
the second equation
and solve
Now substitute y = 2 into
the first equation to solve
for x
x = 2y
3y + 5x = 26
3y + 5(2y) = 26
3y + 10y = 26
13y = 26
so
y=2
x = 2(2)
so
x=4
Therefore the solution point is (4, 2)
Solving Systems by Elimination
-If like terms are lined up in columns, and you see that one of the
variables has opposite coefficients, you may add down the columns in
order to ‘ELIMINATE’ that variable.
EXAMPLE:
Solve
-The y terms eliminate, giving you
an easy equation to solve for x.
2x + 3y = -7
+ _________________
x – 3y = 19
3x
= 12
So
x=4
2(4) + 3y = -7
-Once you solve for x, substitute
that value back into EITHER
equation and solve for y.
8 + 3y = -7
3y = -15
So
y = -5
Answer
(4, -5)
EXAMPLE 2: If no variables will eliminate, then you must multiply one or
both of the equations by an integer that will create opposite coefficients for
one of the variables.
Solve
-No term eliminated, therefore we
had to multiply the first equation by
-2 in order to cancel out the x terms
-When the x terms eliminate, an
easy equation is created to solve
for y.
x + 5y = 2
2x + 3y = -3
 Multiply by -2
-2x – 10y = -4
+ _________________
2x + 3y = -3
-7y = -7
So y = 1
x + 5(1) = 2
-Once you solve for y, substitute
that value into either of the original
equations in order to solve for x.
x+5=2
So x = -3
Answer
(-3, 1)
Section 3-3: Systems of Inequalities
Review
is a dashed boundary line
,
is a solid boundary line
__________
< , >
Shade the appropriate region
above or below the boundary
line.
- In order to graph a system of inequalities, you must graph both
inequalities and determine the OVERLAPPING SHADED REGION
Example: Graph the system
y  x  1
y  2x  3
** If there is no overlap, then there is no solution to the system.