1 Function Find a point of intersection (x, y) of the graphs of functions f and g and you have found a solution of the system y = f(x), y = g(x). |
Solving Systems Using Tables and Graphs (Lesson 3-1)
{
y
=
−
2
x
+
3
y
=
2
x
−
1
left brace . table with 2 rows and 2 columns , row1 column 1 , y equals , column 2 negative 2 x plus 3 , row2 column 1 , y equals , column 2 2 x minus 1 , end table
The solution is (1, 1). |
Systems of Inequalities and Linear Programming (Lessons 3-3 and 3-4)
{
y
>
−
2
x
+
3
y
≤
2
x
−
1
left brace . table with 2 rows and 2 columns , row1 column 1 , y greater than , column 2 negative 2 x plus 3 , row2 column 1 , y less than or equal to , column 2 2 x minus 1 , end table
|
2 Equivalence If the equations of two systems are equivalent, then a solution of the system that is easier to solve is also a solution of the more difficult system. |
Solving Systems Algebraically (Lesson 3-2)
{
−
y
=
−
x
+
2
3
y
=
2
x
−
2
3
(
2
)
=
2
x
−
2
→
→
−
2
y
=
−
2
x
+
4
3
y
=
2
x
−
2
y
=
2
x
=
4
table with 1 row and 3 columns , row1 column 1 , table with 3 rows and 1 column , row1 column 1 , left brace . table with 2 rows and 1 column , row1 column 1 , negative y equals negative x plus 2 , row2 column 1 , 3 y equals 2 x minus 2 , end table , row2 column 1 , , row3 column 1 , 3 open 2 close equals 2 x minus 2 , end table , column 2 table with 3 rows and 1 column , row1 column 1 , rightwards arrow , row2 column 1 , , row3 column 1 , rightwards arrow , end table , column 3 table with 4 rows and 3 columns , row1 column 1 , negative 2 y , column 2 equals , column 3 negative 2 x plus 4 , row2 column 1 , 3 y , column 2 equals , column 3 2 x minus 2 , row3 column 1 , y , column 2 equals , column 3 2 , row4 column 1 , x , column 2 equals , column 3 4 , end table , end table
The solution is x = 4, y = 2 |
Systems With Three Variables (Lesson 3-5)
{
−
2
x
+
y
+
z
=
−
3
2
x
−
y
+
z
=
−
1
−
2
x
,
−
y
−
z
=
−
1
left brace . table with 3 rows and 4 columns , row1 column 1 , negative 2 x plus , column 2 y plus , column 3 z equals , column 4 negative 3 , row2 column 1 , 2 x minus , column 2 y plus , column 3 z equals , column 4 negative 1 , row3 column 1 , negative 2 x comma minus , column 2 y minus , column 3 z equals , column 4 negative 1 , end table
x = 1, y = 1,
z
=
−
2
z equals negative 2
|
3 Solving Equations and Inequalities The matrix row operations of adding rows and multiplying a row by a constant are equivalent to addition and multiplication properties of equality. |
Solving Systems Using Matrices (Lesson 3-6)
[
−
2
3
1
2
−
1
1
]
[
1
0
1
0
1
1
]
→
x
=
1
,
y
=
1
table with 2 rows and 1 column , row1 column 1 , . matrix with 2 rows and 3 columns , row1 column 1 , negative 2 , column 2 3 , column 3 1 , row2 column 1 , 2 , column 2 negative 1 , column 3 1 , end matrix , row2 column 1 , left bracket . table with 2 rows and 3 columns , row1 column 1 , 1 , column 2 0 , column 3 1 , row2 column 1 , 0 , column 2 1 , column 3 1 , end table . right bracket rightwards arrow x equals 1 comma y equals 1 , end table
|