256 CHAPTER 4 Systems of Linear Equations
To find x, substitute 2 for yin equation (4).
Let in (4).
Multiply.
Subtract 6.
Divide by 2.
Substitute 3 for xand 2 for yin working equation (2).
(2)A check of the solution confirms that the solution set is
51 3, 2, 1 26.1 3, 2, 1 2
z= 13 + 2 +z= 6x+y+z= 6x= 32 x= 62 x+ 6 = 122 x+ 3122 = 12 y= 2CONCEPTS EXAMPLES
Step 5 Find the value of the remaining variable.
Step 6 Check the ordered-triple solution in each of the
original equations of the system. Then write the
solution set.
4.3 Applications of Systems of Linear
Equations
Use the six-step problem-solving method.
Step 1 Read the problem carefully.
Step 2 Assign variables.
Step 3 Write a system of equations that relates the
unknowns.
Step 4 Solve the system.
Step 5 State the answer.
Step 6 Check.
The perimeter of a rectangle is 18 ft. The length is 3 ft more than
twice the width. What are the dimensions of the rectangle?
Let xrepresent the length and yrepresent the width. From the
perimeter formula, one equation is From the problem,
another equation is Solve the systemto get and The length is 7 ft, and the width is 2 ft. Since
the perimeter is
and
the solution checks.2172 + 2122 =18, 2122 + 3 =7,
x= 7 y=2.x= 2 y+ 32 x+ 2 y= 18x= 2 y+3.2 x+ 2 y=18.4.4 Solving Systems of Linear Equations
by Matrix Methods
Matrix Row Operations
1.Any two rows of the matrix may be interchanged.
2.The elements of any row may be multiplied by any
nonzero real number.
becomes
Interchange
andbecomes C^13 R 21
1
0
5
3
6
7
-^23
4
C S
1
3
0
5
9
6
7
- 2
4
S
C R 1 R 2.
3
1
0
9
5
6
- 2
7
4
C S
1
3
0
5
9
6
7
- 2
4
S
(continued)3.Any row may be changed by adding to the elements of
the row the product of a real number and the elements
of another row.
becomes C -3R 1 +R 2