THE SOLUTION OF SIMULTANEOUS EQUATIONS BY MATRICES AND DETERMINANTS 279F
Now try the following exercise.
Exercise 113 Further problems on solving
simultaneous equations using matricesIn Problems 1 to 5 usematricesto solve the
simultaneous equations given.- 3x+ 4 y= 0
2 x+ 5 y+ 7 =0[x=4,y=−3]- 2p+ 5 q+ 14. 6 = 0
3. 1 p+ 1. 7 q+ 2. 06 = 0
[p= 1 .2,q=− 3 .4]
3.x+ 2 y+ 3 z= 5
2 x− 3 y−z= 3
− 3 x+ 4 y+ 5 z= 3
[x=1,y=−1,z=2]- 3a+ 4 b− 3 c= 2
− 2 a+ 2 b+ 2 c= 15
7 a− 5 b+ 4 c= 26
[a= 2 .5,b= 3 .5,c= 6 .5]
5.p+ 2 q+ 3 r+ 7. 8 = 0
2 p+ 5 q−r− 1. 4 = 0
5 p−q+ 7 r− 3. 5 = 0
[p= 4 .1,q=− 1 .9,r=− 2 .7]- In two closed loops of an electrical cir-
cuit, the currents flowing are given by the
simultaneous equations:
I 1 + 2 I 2 + 4 = 0
5 I 1 + 3 I 2 − 1 = 0Use matrices to solve forI 1 andI 2.
[I 1 =2,I 2 =−3]- The relationship between the displacement,
s, velocity,v, and acceleration,a, of a piston
is given by the equations:
s+ 2 v+ 2 a= 4
3 s−v+ 4 a= 25
3 s+ 2 v−a=− 4Use matrices to determine the values ofs,v
anda.
[s=2,v=−3,a=4]- In a mechanical system, acceleration x ̈,
velocityx ̇and distancexare related by the
simultaneous equations:
3. 4 x ̈+ 7. 0 x ̇− 13. 2 x=− 11. 39− 6. 0 x ̈+ 4. 0 ̇x+ 3. 5 x= 4. 982. 7 x ̈+ 6. 0 x ̇+ 7. 1 x= 15. 91Use matrices to find the values ofx ̈,x ̇andx.[ ̈x= 0 .5,x ̇= 0 .77,x= 1 .4]26.2 Solution of simultaneous
equations by determinants(a) When solving linear simultaneous equations in
two unknowns using determinants:
(i) write the equations in the forma 1 x+b 1 y+c 1 = 0a 2 x+b 2 y+c 2 = 0and then(ii) the solution is given by
x
Dx=−y
Dy=1
Dwhere Dx=∣
∣
∣
∣b 1 c 1
b 2 c 2∣
∣
∣
∣i.e. the determinant of the coefficients left
when thex-column is covered up,Dy=∣
∣
∣
∣a 1 c 1
a 2 c 2∣
∣
∣
∣i.e. the determinant of the coefficients left
when they-column is covered up,and D=∣
∣
∣
∣a 1 b 1
a 2 b 2∣
∣
∣
∣i.e. the determinant of the coefficients left
when the constants-column is covered up.Problem 3. Solve the following simultaneous
equations using determinants:3 x− 4 y= 127 x+ 5 y= 6. 5