Basic Engineering Mathematics, Fifth Edition

Solving simultaneous equations 97

Problem 13. The equation of a straight line,

of gradientmand intercept on they-axisc,is

y=mx+c. If a straight line passes through the

point wherex=1andy=−2, and also through

the point wherex= 3 .5andy= 10 .5, find the

values of the gradient and they-axis intercept

Substitutingx=1andy=−2intoy=mx+cgives

− 2 =m+c (1)

Substitutingx= 3 .5andy= 10 .5intoy=mx+c
gives

10. 5 = 3. 5 m+c (2)

Subtracting equation (1) from equation (2) gives

12. 5 = 2. 5 m, from which, m=

12. 5

2. 5

= 5

Substitutingm=5 into equation (1) gives

− 2 = 5 +c

c=− 2 − 5 =− 7

Checking, substitutingm=5andc=−7inequation
(2), gives

RHS=( 3. 5 )( 5 )+(− 7 )= 17. 5 − 7 = 10. 5 =LHS

Hence, thegradientm= 5 and they-axis intercept
c=− 7.

Problem 14. When Kirchhoff’s laws are applied

to the electrical circuit shown in Figure 13.1, the

currentsI 1 andI 2 are connected by the equations

27 = 1. 5 I 1 + 8 (I 1 −I 2 ) (1)

− 26 = 2 I 2 − 8 (I 1 −I 2 ) (2)

27 V 26 V

8 V

1.5V^2 V

l 1 l 2

(l 12 l 2 )

Figure 13.1

Solve the equations to find the values of currentsI 1

andI 2

Removing the brackets from equation (1) gives

27 = 1. 5 I 1 + 8 I 1 − 8 I 2

Rearranging gives

9. 5 I 1 − 8 I 2 = 27 (3)

Removing the brackets from equation (2) gives

− 26 = 2 I 2 − 8 I 1 + 8 I 2

Rearranging gives

− 8 I 1 + 10 I 2 =− 26 (4)

Multiplying equation (3) by 5 gives

47. 5 I 1 − 40 I 2 = 135 (5)

Multiplying equation (4) by 4 gives

− 32 I 1 + 40 I 2 =− 104 (6)

Adding equations (5) and (6) gives

15. 5 I 1 + 0 = 31

I 1 =

31

15. 5

= 2

SubstitutingI 1 =2 into equation (3) gives

9. 5 ( 2 )− 8 I 1 = 27

19 − 8 I 2 = 27

19 − 27 = 8 I 2

− 8 = 8 I 2

and I 2 =− 1

Hence, the solution isI 1 = 2 andI 2 =− 1 (which may

be checked in the original equations).

Problem 15. The distancesmetres from a fixed

point of a vehicle travelling in a straight line with

constant acceleration,am/s^2 ,isgivenby

s=ut+^12 at^2 ,whereuis the initial velocity in m/s

andtthe time in seconds. Determine the initial

velocity and the acceleration given thats=42m

whent=2s,ands=144m whent=4s.Alsofind

the distance travelled after 3s