In
late November, the jet stream
moving across North America could be described by the quadratic
equation 
, where Chicago
was at the origin.
Suppose a plane’s
route is described by the linear equation 
. What are the coordinates of the point at which
turbulence will
occur? This
problem will be solved in Example 7.
Like a linear system of
equations, the solution of a quadratic-linear system of equations is the ordered pair
that satisfies both equations. A quadratic-linear system can have 0, 1, or 2
solutions, as shown below.
You can solve
quadratic-linear systems of equations by using some of the methods you used for
solving systems of linear equations. One method is graphing.
Determine whether each
system of equations has one solution, two solutions, or no solution
by graphing. If the system has one solution or two solutions, name them.
1. y =
x2
y = x + 2
Alternative Solutions:
The graphs appear to intersect at (–1,
1) and (2, 4). Check this estimate by substituting the coordinates into each
equation.
Check:
Check that the ordered pair (2, 4) satisfies both equations. The solutions of the
system of equations are (–1, 1) and (2, 4).
2. y = 2x2
+ 5
y = –x + 3
Alternative Solutions:
Because the graphs do not intersect, there is
no solution to this system of equations.
3. y = –x2 + 3
y = 2x + 4
Alternative Solutions:
The graphs appear to intersect at (–1,
2).
Check:
The
solution of the system of equations is (–1, 2).
You can also solve
quadratic-linear systems of equations by using the substitution method.
Example
4. Use
substitution to solve each system of equations.
y = –4
y = x2 – 2
Alternative Solutions:
Substitute 4 for y in the second
equation. Then solve for x.
The solution of the system of equations is
(0, –4).
Check:
Sketch the graphs of the equations. The parabola and line
appear to intersect at (0, –4). The solution is correct.
5. y = x2
y = –3
Alternative Solutions:
Substitute
–3 for y in the second equation. Then solve for x.
There is no real
solution because the square root of a negative number is not a real number. Check by graphing.
6. y = x2 – 4x
+ 6
y = –x + 4
Alternative Solutions:
Substitute –x + 4 for y in the
second equation. Then solve for x.
Substitute the values of x in either equation to find the
corresponding values of y. Choose the equation that is easier for you to
solve.
The solutions of the
system of equations are (2, 2) and (1, 3). The graph shows that the solutions
are probably correct. You can also check by
substituting the ordered pairs into the original equations.
Meteorology
Link
7. Refer
to the application at the beginning of the lesson. What are the coordinates of
the point at which turbulence will occur?
Alternative Solutions:
Use substitution to solve the system 
and 
. Substitute 
for y in
the first equation.
Substitute the values of x to find the
corresponding values of y.
The solutions of the system are (–8, 4) and
(6, –3). This means that turbulence will occur as the plane passes through
points having these coordinates. Check
by substituting the coordinates into each equation and by looking at the graph
at the beginning of the lesson.
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