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3
Big Idea 1
TEKS
Big Idea 2
TEKS
Big Idea 3
TEKS
2A.3.B
2A.3.A
a.2
BIG IDEAS For Your Notebook
Solving Systems of Equations Using a Variety of Methods
Method When to use
Graphing: Graph each equation in the
system. A point where the graphs intersect
is a solution.
Substitution: Solve one equation for one
of the variables and substitute into the
other equation(s).
Elimination: Multiply equations by
constants, then add the revised equations
to eliminate a variable.
Cramer’s rule: Use determinants to find
the solution.
Inverse matrices: Write the system as a
matrix equation AX 5 B. Multiply each side
by A 21 on the left to obtain the solution
X 5 A 21 B.
Graphing Systems of Equations and Inequalities
System of equations
with 1 solution
y
x
System of equations
with many solutions
y
x
The equations have only two variables and
are given in a form that is easy to graph.
One of the variables in the system has a
coefficient of 1 or 21.
None of the variables in the system have a
coefficient of 1 or 21.
The determinant of the coefficient matrix is
not zero.
The determinant of the coefficient matrix is
not zero.
System of equations
with no solution
y
x
System of
inequalities
Intersecting lines Coinciding lines Parallel lines Shaded region
Using Matrices
Addition, subtraction, and
scalar multiplication
Fa b e f a 1 e b1f c dG1Fg hG5Fc1g d1hG Fa b e f a 2 e b2f c dG2Fg hG5Fc2g d2hG a b ka kb
kF c dG5F kc kdG
Matrix
multiplication
Fa b
e f
5
c dGFg hG
Fae 1 bg af 1 bh
ce 1 dg cf 1 dhG
Inverse matrices
a b
If A 5F then
c dG,
A 21 5 1 d 2b
}
or
2c aG
A 21 1 d 2b
5}
ad 2 cbF2c aG.
⏐A⏐ F
y
x
Chapter Summary 221