The equation is defined to be a mathematical statement of equality. If the equality is true for certain value of the variable involved, the equation is often called a conditional equation and equality sign ‘=’ is used .
- LINEAR (Simple) EQUATION : An equation in which highest power of the variable is 1 is called a linear equation. For e.g ax+b=0.
- SIMULTANEOUS LINEAR EQUATION: Two or more linear equations involving two or more variable are called simultaneous linear equations. For e.g. ax+by+c=0.
- QUADRATIC EQUATIONS: an equation of degree 2 ( highest power of the variable is 2 ) is called quadratic equation. For e.g. ax²+ bx +c =0
- CUBIC EQUATION: the equation of degree 3 is called cubic equation. For e.g. ax³+ bx²+cx+d =0
Results
#1. The equation - 7x + 1 = 5 - 3x will be satisfied for x equal to:
solution -7x 1 =5 -3x
by transposing the variables in one side and the constants in other side we have
-7x 3x = 5-1
-4x = 4
x= 4 / -4
x = -1 Answer
#2. The root of the equation x+4 /4+x-5/3 =11 is
solution x 4 / 4 X-5 / 3 = 11
cross multiply
3(x 4) 4(x-5)=11*12
3x 12 4x-20=132
by transposing the variables in one side and the constants in other side we have .
3x 4x-20 12=132
7x – 8=132
7x=132 8
7x=140
x=140/7
x=20 Answer.
#3. The equation x+y=14 ; x-y =4
solution: x y=14 ; x-y =4
cross multiply
(x y=14)……………. (a)
(x-y=4)………………. (b)
by making (a)1(x y=14)
(b)1(x-y=4)
sign change we get ….. 2y=10
y= 10/2
y=5
put y=5 in first equation
x y=14
x 5=14
x=14-5
x=9; y=5 answer
#4. The solution of the set of equations 3x+4y=7,4x-y=3
solution : 3x 4y=7,4x-y=3
3x 4y=7………..multiply by 4
4x-y=3………..multiply by 3
we get
12x 16y=28
12x -3y=9 …….sign change
19y=19
y= 19/19
y=1
put in first equation 3x 4y=7
3x 4(1)=7
3x=7-4
3x=3
x=3/3
x=1;y=1
#5. The simultaneous equations 7x-3y=31,9x-5y=41
#6. The fourth parts of a number exceeds the sixth part by 4 . The number is
solution : the fourth parts of a number exceeds the sixth part by 4 . The number is
x/4=x/6 4………….. cross multiply right side
x/4=x 24/6………… cross multiply both side
x(6)=4(x 24)
6x=4x 96
6x-4x=96
2x=96
x=96/2
x=48 answer
#7. The sum of two numbers is 52 and their difference is 2 . The numbers are
#8. Divide 25 into two parts so that sum of their reciprocals is 1/6
by the question 1/x 1/25-x=1/6
25-x x/x(25-x)=1/6
150=25x-x²
x²-25x 150=0
x²-15x-10x 150=0
x(x-15)-10(x-15)=0
(x-15)(x-10)=0
x=10,15 answer
#9. If the roots of the equation 2x²+8x-m³=0 are equal then value of m is
solution: 2x² 8x-m³=0
ax² bx c=0
a=2,b=8,c=-m³
b²-4ac=0
8²-4*2*(-m³)=0
64-8(-m³)=0
64 8m³=0
m³=(-8)³
m=(-2)answer
#10. If αβ be the roots of the equation 2x²-4x -3x =0 the value of α² + β² is
#11. The values of x for the equation x²+9x+18=6-4x are
#12. Find the roots of the quadratic equation √2x²+7x+5√2=0
√2x² 7x 5√2=0……….(7x factor )
√2x² 5x 2x 5√2=0
x(√2x 5) √2(√2x 5)=0
(x √2)(√2x 5)=0
The roots of √2x² 7x 5√2=0 are the values of x for which (x √2)(√2x 5)=0
There for x √2=0 √2x 5=0
x=-√2 x=-5/√2
#13. The rational root of the equation 2x³-x²-4x+2=0 is
#14. The roots of x³+x²-x-1 are
#15. Solving x³-6x²+11x-6=0 we get the following roots
x³-6x² 11x-6=0
1 is a root by sight,Dividing the polynomial by (x-1)
(x-1)(x²-5x 6)=0
(x-1)(x-2)(x-3)=0
roots: 1,2,3
#16. Solve x³-7x+6=0
x³-7x 6=0 …put x-1
x³-x² x²-x-6x 6=0
x²(x-1) x(x-1)-6(x-1)=0
(x-1)(x² x-6)=0
(x-1)(x² 3x-3x-6)=0
(x-1){x(x 3)-2(x 3)}=0
(x-1)(x-2)(x 3)
x=1,2,-3
#17. Solve the equation 4x/3-1=14/15x+19/5
4x/3-14x/15=19/5 1
(20-14)x/15 =19 5/5
6x/15=24/5
x=24×15/5×6
x=12
#18. If a number of which the half is greater than 1/5th of the number by 15 then the number is
assumed number is x
half of number =x/2
According to the problem given,
x/2-x/5=15
5x-2x/10=15
3x/10=15
x=15*10/3
x=5*10
x=50