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5 Solve the following pair of equations as a two-by-two system, including the complexnumber solutions, if any Let x be the independent variable, and then define y as a function of x in each equation: x 2 + x y = 1 and x 2 2x y = 2 6 Check the solution(s) to Prob 5 in the original equations for correctness 7 Solve the following pair of equations as a two-by-two system, including the complexnumber solutions, if any Let x be the independent variable, and then define y as a function of x in each equation: x2 + y = 0 and 2x 3 y = 0 Explain why one of the solutions has multiplicity 2 8 Check the solution(s) to Prob 7 in the original equations for correctness 9 Solve the following pair of equations as a two-by-two system, including the complexnumber solutions, if any Let x be the independent variable, and then define y as a function of x in each equation: 4x 3 + 2x 2 + 2x 2y 8 = 0 and 3x 3 2x 2 + 4x y 5 = 0 What is the multiplicity of each solution 10 Check the solution(s) to Prob 9 in the original equations for correctness

Cable signals can be modulated using any of the techniques outlined earlier in this chapter The most familiar example is cable television

In this chapter, we ll graph the systems we solved in Chap 27 to see how they work in the realm of real numbers Real solutions always appear as intersection points between the graphs of functions However, when a two-by-two system has non-real complex solutions, those solutions don t show up as intersections between real-number graphs

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All radio and TV signals are electromagnetic waves The radio or TV transmitter output is coupled into an antenna system located at some distance from the transmitter The energy follows a transmission line, also called a feed line, from the transmitter PA output to the antenna itself Most transmission lines are coaxial cables There are other types, used in special applications At microwaves, hollow tubes called waveguides are used to transfer the energy from a transmitter to the antenna A waveguide is more efficient than coaxial cable at the shortest radio wavelengths

Bits, Bytes, and Baud A bit is almost always represented by either 0 or 1 A group of 8 bits is a called an octet, and in many systems this also corresponds to a unit called a byte Large quantities of data can be expressed either according to powers of 2, or according to powers of 10 This can cause some confusion and gives rise to endless debates over semantics One kilobit (kb) is equal to 1000 bits A megabit (Mb) is 1000 kilobits, or 1,000,000 bits A gigabit (Gb) is 1000 megabits, or 1,000,000,000 bits When data is expressed in bits, powers of 10 are used to define large quantities If you hear about a modem that operates at 56 kbps, it means 56,000 bits per second (bps) Bits, kilobits, megabits, and gigabits per second (bps, kbps, Mbps, and Gbps) are commonly used to express data in communications

The first example we worked out in Chap 27 involved a system of two equations, one linear and the other quadratic: 2x y + 1 = 0 and y = x2 + x 5 We called x the independent variable, so we rewrote the first equation as a function of x The two-function system then became y = 2x + 1 and y = x2 + x 5

First, we tabulate some points The graphs we re about to create aren t meant to be precise We aren t trying to pinpoint the real solutions using the graphs; we ve already calculated them! The graphs serve only to geometrically illustrate the functions and their solutions

494 Data transmission Radio amateurs sometimes use parallel-wire transmission lines, resembling the ribbon cable popular for use with consumer TV receiving antennas In a parallel-wire line, the RF currents in the two conductors are always 180 degrees out of phase, so that their electromagnetic fields cancel each other This keeps the transmission line from radiating, guiding the EM field along toward the antenna The energy is radiated when it reaches the antenna The radio frequency bands are generally categorized from very low frequency (VLF) through microwaves, according to the breakdown in Table 26-1 These waves propagate through the atmosphere, or through space, in different ways depending on their frequency Radio signal propagation is discussed in the next chapter

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