What is solution ?
The value which satisfies the given equation is known as solution. There are three types of solutions.
(i) Unique solution
(ii) Infinitely many solution
(iii) No solution
What is unique solution ?
Exactly one value will satisfy the equation. That particular value is the solution of the given equation.
For example,
Problem 1 :
2x + 4 = 9 + x
Solution :
2x + 4 = 9 + x
Subtract x from each side of the equation.
2x + 4 – x = 9 + x – x
Combine the like terms.
x + 4 = 9
Add 4 to each side of the equation.
x + 4 – 4 = 9 – 4
x = 5
So, it is one solution.
What is infinitely solution ?
Infinite values will satisfy the given equation.
Problem 2 :
5a + 5 = 5(a + 1)
Solution :
5a + 5 = 5(a + 1)
5a + 5 = 5a + 5
Subtract 5 from each side of the equation.
5a + 5 - 5 = 5a + 5 – 5
5a = 5a
So, it is infinitely many solutions.
What is no solution ?
No values will satisfy the given equation.
Problem 3 :
5a + 5 = 5(a + 2)
Solution :
5a + 5 = 5(a + 2)
5a + 5 = 5a + 10
No values of a will satisfy the equation given above. So, there is no solution.
Solve each equation. Indicate if an equation has no solution or infinitely many solutions.
Problem 4 :
b + 3b – 10 = 2(2b – 5)
Solution :
b + 3b – 10 = 2(2b – 5)
4b – 10 = 4b – 10
Add 10 to each side of the equation.
4b – 10 + 10 = 4b – 10 + 10
4b = 4b
So, it is infinitely many solutions.
Problem 5 :
y + 5y – 6 = 3(2y – 2)
Solution :
y + 5y – 6 = 3(2y – 2)
6y – 6 = 6y – 6
Add 6 to each side of the equation.
6y – 6 + 6 = 6y – 6 + 6
6y = 6y
So, it is infinitely many solutions.
Problem 6 :
-4i – 11 = 4i/2 + 7
Solution :
-4i – 11 = 4i/2 + 7
-4i – 11 = 2i + 7
Subtract 2i from each side of the equation.
-4i – 11 – 2i = 2i + 7 – 2i
-6i – 11 = 7
Add 11 to each side of the equation.
-6i – 11 + 11 = 7 + 11
-6i = 18
Divide each side by 6.
-6i/6 = 18/6
-i = 3
i = -3
So, it is one solution.
Problem 7 :
-4z + 6 = 4(-z + 2)
Solution :
-4z + 6 = 4(-z + 2)
-4z + 6 = -4z + 8
Add 4z to each side of the equation.
-4z + 6 + 4z = -4z + 8 + 4z
6 = 8
So, it is no solution.
Problem 8 :
4i + 3 = 4(i + 1)
Solution :
4i + 3 = 4(i + 1)
4i + 3 = 4i + 4
Subtract 4i from each side of the equation.
4i + 3 – 4i = 4i + 4 – 4i
3 = 4
So, it is no solution.
Problem 9 :
6z + z = 5z + 10
Solution :
6z + z = 5z + 10
7z = 5z + 10
Subtract 5z from each side of the equation.
7z – 5z = 5z – 5z + 10
2z = 10
Divide each side by 2.
2z/2 = 10/2
z = 5
So, it is one solution.
Problem 10 :
2c – c = 3c - 5
Solution :
2c – c = 3c - 5
c = 3c – 5
Subtract 3c from each side of the equation.
c – 3c = 3c – 3c – 5
-2c = -5
Divide each side by 2.
-2c/2 = -5/2
c = -5/2
So, it is one solution.
Problem 11 :
12n + 7 = 4n + 8n + 6
Solution :
12n + 7 = 4n + 8n + 6
12n + 7 = 12n + 6
Subtract 12n from each side of the equation.
12n + 7 – 12n = 12n + 6 – 12n
7 = 6
So, it is no solution.
Problem 12 :
12d(3 – 2) = 24
Solution :
12d(3 – 2) = 24
36d – 24d = 24
12d = 24
Divide each side by 12.
12d/12 = 24/12
d = 2
Problem 13 :
4x = x + 10.5
Solution :
4x = x + 10.5
Subtract x from each side of the equation.
4x - x = x - x + 10.5
3x = 10.5
Divide each side by 3.
3x/3 = 10.5/3
x = 3.5
So, it is one solution.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM