Linear equation in one variable will be in the form
ax + b = c
Since the highest exponent of x is 1, it is named as linear.
To solve linear equation in one variable, we follow the steps given below.
Step 1 :
Collect the variable term in one side. If it is necessary, we may have to use distributive property.
Step 2 :
Combine the numerical value to the other side. Using inverse operation, we can do this.
Step 3 :
The value that satisfies the given equation is the solution.
Solve the equations. Write null or all where appropriate. Show fractions in simplest form.
Problem 1 :
5(y + 2) = 20
Solution :
5(y + 2) = 20
Use distributive property.
5y + 10 = 20
Subtract 10 from both sides.
5y + 10 – 10 = 20 – 10
5y = 10
Divide both sides by 5.
5y/5 = 10/5
y = 2
It has unique solution.
Problem 2 :
2(x – 4) = 2x – 8
Solution :
2(x – 4) = 2x – 8
Use distributive property.
2x – 8 = 2x – 8
Add 8 to both sides.
2x – 8 + 8 = 2x – 8 + 8
2x = 2x
Divide both sides by 2.
2x/2 = 2x/2
x = x
So, the solution is true for all numbers.
Problem 3 :
6(k + 3) = 2(4k + 5)
Solution :
6(k + 3) = 2(4k + 5)
Use distributive property on both sides.
6(k + 3) = 2(4k + 5)
6k + 18 = 8k + 10
Subtract 8k from both sides.
6k + 18 – 8k = 8k – 8k + 10
6k + 18 – 8k = 10
Subtract 18 from both sides.
6k + 18 – 8k - 18 = 10 -18
-2k = -8
Divide both sides by -2.
-2k/-2 = -8/-2
k = 4
So, it has one solution.
Problem 4 :
30 – 2x = 2(3x + 3)
Solution :
30 – 2x = 2(3x + 3)
Use distributive property.
30 – 2x = 6x + 6
Subtract 6x from both sides.
30 – 2x – 6x = 6x – 6x + 6
30 – 8x = 6
Subtract 30 from both sides.
30 – 8x - 30 = 6 – 30
-8x = -24
Divide both sides by -8.
-8x/-8 = -24/-8
x = 3
So, it has one solution.
Problem 5 :
16 = 4(n – 5) + 4
Solution :
16 = 4(n – 5) + 4
16 = 4n – 20 + 4
16 = 4n – 16
Add 16 to both sides.
16 + 16 = 4n – 16 + 16
32 = 4n
Divide both sides by 4.
32/4 = 4/4n
8 = n
So, it has one solution.
Problem 6 :
7p + 9 = 4(p – 3)
Solution :
7p + 9 = 4(p – 3)
Use distributive property.
7p + 9 = 4p – 12
Add 12 to both sides.
7p + 9 + 12 = 4p – 12 + 12
7p + 21 = 4p
Subtract 7p from both sides.
7p – 7p + 21 = 4p – 7p
21 = -3p
Divide both sides by -3.
-21/3 = -3p/-3
-7 = p
So, it has one solution.
Problem 7 :
3(2m + 4) = 6m + 15
Solution :
3(2m + 4) = 6m + 15
Use distributive property.
6m + 12 = 6m + 15
Subtract 6m from both sides.
6m – 6m + 12 = 6m – 6m + 15
12 = 15
So, it has null solution.
Problem 8 :
2(4w + 7) = 3(6 + 2w)
Solution :
2(4w + 7) = 3(6 + 2w)
Use distributive property on both sides.
8w + 14 = 18 + 6w
Subtract 14 from both sides.
8w + 14 - 14 = 18 + 6w – 14
8w = 4 + 6w
Subtract 6w from both sides.
8w – 6w = 4 + 6w – 6w
2w = 4
Divide both sides by 2.
2w/2 = 4/2
w = 2
So, it has one solution.
Problem 9 :
25 = 6(c + 7) - 14
Solution :
25 = 6(c + 7) - 14
25 = 6c + 42 – 14
25 = 6c + 28
Subtract 28 from both sides.
25 – 28 = 6c + 28 – 28
-3 = 6c
Divide both sides by 6.
-3/6 = 6c/6
-1/2 = c
So, it has one solution.
Problem 10 :
3(3h + h) = 2(h + 5)
Solution :
3(3h + h) = 2(h + 5)
Use distributive property on both sides.
9h + 3h = 2h + 10
12h = 2h + 10
Subtract 2h from both sides.
12h – 2h = 2h – 2h + 10
10h = 10
Divide both sides by 10.
10h/10 = 10/10
h = 1
So, it has one solution.
Problem 11 :
3(2 + n) = 10 + 2n
Solution :
3(2 + n) = 10 + 2n
Use distributive property.
6 + 3n = 10 + 2n
Subtract 6 from both sides.
6 + 3n - 6 = 10 + 2n – 6
3n = 4 + 2n
Subtract 2n from both sides.
3n - 2n = 4 + 2n – 2n
n = 4
So, it has one solution.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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