Basic Engineering Mathematics, Fifth Edition

Solving simultaneous equations 99

Whenc= 52 , T=100, hence

52 =a+ 100 b (1)

Whenc= 172 , T=400, hence

172 =a+ 400 b (2)

Equation (2) – equation (1) gives

120 = 300 b

from which, b=^120
300

= 0. 4

Substitutingb= 0 .4 in equation (1) gives

52 =a+ 100 ( 0. 4 )

a= 52 − 40 = 12

Hence,a= 12 and b= 0. 4

Now try the following Practice Exercise

PracticeExercise 52 Practical problems

involving simultaneous equations (answers

on page 345)

1. In a system of pulleys, the effortPrequired to
   raisealoadWis given byP=aW+b,where
   aandbare constants. IfW=40 whenP= 12
   andW=90 whenP=22, find the values of
   aandb.


2. Applying Kirchhoff’s laws to an electrical
   circuit produces the following equations:

5 = 0. 2 I 1 + 2 (I 1 −I 2 )

12 = 3 I 2 + 0. 4 I 2 − 2 (I 1 −I 2 )

Determine the values of currentsI 1 andI 2

3. Velocityvis given by the formulav=u+at.
   Ifv=20 whent=2andv=40 whent=7,
   find the values ofuanda. Then, find the
   velocity whent= 3. 5


4. Three new cars and 4 new vans supplied to a
   dealer together cost £97700 and 5 new cars
   and 2 new vans of the same models cost
   £103100. Find the respective costs of a car
   and a van.


5. y=mx+cis the equation of a straight line
   of slopemandy-axis interceptc. If the line
   passes through the point wherex=2and

y=2, and also through the point wherex= 5

andy= 0 .5,findtheslopeandy-axis intercept

of the straight line.

6. The resistanceRohms of copper wire att◦C
   is given byR=R 0 ( 1 +αt),whereR 0 is the
   resistanceat 0◦Candαisthetemperaturecoef-
   ficient of resistance. IfR= 25. 44 at 30◦C
   andR= 32. 17 at 100◦C, findαandR 0


7. The molar heat capacity of a solid compound
   is given by the equationc=a+bT.When
   c= 60 ,T=100 and whenc= 210 ,T=400.
   Find the values ofaandb.


8. In an engineering process, two variablespand
   qare related byq=ap+b/p,whereaandb
   are constants. Evaluateaandbifq=13 when
   p=2andq=22 whenp=5.


9. In a system of forces, the relationshipbetween
   two forcesF 1 andF 2 is given by

5 F 1 + 3 F 2 + 6 = 0

3 F 1 + 5 F 2 + 18 = 0

Solve forF 1 andF 2

13.6 Solving simultaneous equations

in three unknowns

Equations containing three unknowns may be solved

using exactly the same procedures as those used with

two equations and two unknowns, providing that there

are three equations to work with. The method is demon-

strated in the following worked problem.

Problem 18. Solve the simultaneous equations.

x+y+z=4(1)

2 x− 3 y+ 4 z= 33 (2)

3 x− 2 y− 2 z=2(3)

There are a number of ways of solving these equations.

One method is shown below.

The initial object is to produce two equations with two

unknowns. For example, multiplying equation (1) by 4

and then subtracting this new equationfrom equation(2)

will produce an equation with onlyxandyinvolved.