College Algebra 9th txtbk

4 CHAPTER 7 SYSTEMS OF EQUATIONS AND MATRICES
Z Figure 3
19 inches
x
y
This system is solved using the procedures outlined in Example 2. However, in this case,
we are only interested in real solutions. We start by solving the second equation for y in
terms of x and substituting the result into the first equation.
y 175
x
x 2 1752
x 2 19 2
x 4 30,625 361x 2
x 4 361x 2 30,625 0
Multiply both sides by x 2 and simplify.
Subtract 361x 2 from each side.
Quadratic in x 2
Solve the last equation for x 2 using the quadratic formula, then solve for x:
361 2361
x 2 4(1)(30,625)
B 2
15.0 inches or 11.7 inches
Substitute each choice of x into y 175x to find the corresponding y values:
For x 15.0 inches,
For x 11.7 inches,
y 175
y 175 15.0 inches
15 11.7 inches 11.7
In either case, the dimensions are 15.0 by 11.7 inches.
MATCHED PROBLEM 3
An engineer is hired to design a rectangular television screen with a 21-inch diagonal and
a 209-square-inch area. Find the dimensions of the screen to the nearest tenth of an inch.
Technology Connections
In Example 3, we’re only concerned with real solutions, so
graphic techniques can also be used to approximate the solutions
(see Fig. 4). As we saw in Section 2-2, graphing a circle
on a graphing calculator requires two functions, one for
the upper half of the circle and another for the lower half.
[Note: since x and y must be nonnegative real numbers, we
ignore the intersection points in the third quadrant—see
Fig 4(a).]
40
16
16
60
60
8
18
8
18
40
(a) y 1 2361 x 2
y 2 2361 x 2
y 3 175
x
10
(b) Intersection point:
(11.7, 15.0)
10
(c) Intersection point:
(15.0, 11.7)
Z Figure 4 Graphic solution of x 2 y 2 19 2 , xy 175.