To solve an equation means to find all values of the variable that make the equation a true statement. One way to do this is to isolate the variable that has a coefficient of 1 onto one side of the equation. You can do this using the rules of algebra called properties of equality.
Addition Property :
If a = b, then
a + c = b + c
Example :
If x - 3 = 5, then
(x - 3) + 3 = (5) + 3
Subtraction Property :
If a = b, then
a - c = b - c
Example :
If x + 3 = 5, then
(x + 3) - 3 = (5) - 3
Multiplication Property :
If a = b, then
c ⋅ a = c ⋅ b
Example :
If x/2 = 7, then
2 ⋅ (x/2) = 2 ⋅ 7
Division Property :
If a = b and c ≠ 0, then
a/c = b/c
Example :
If 3x = 15, then
3x/5 = 15/5
Problem 1 :
Solve for a :
a - (-10) = -12
Solution :
a - (-10) = -12
a + 10 = -12
Subtract 10 from both sides.
a = -22
Problem 2 :
Solve for x :
-24 = 8x
Solution :
-24 = 8x
Divide both sides by 8.
-3 = x
Problem 3 :
-10 + x = 12
Given the above equation, what is the value of 2 - (5 - x)?
Solution :
-10 + x = 12
Add 10 to both sides.
x = 22
Substitute x = 22 in 2 - (5 - x).
= 2 - (5 - 22)
= 2 - (-17)
= 2 + 17
= 19
Problem 4 :
If 33 - y = y + 27 - 5y, what is the value of 33 + 3y?
Solution :
33 - y = y + 27 - 5y
Simplify.
33 - y = 27 - 4y
Add 4y to both sides.
33 + 3y = 27
Problem 5 :
If x/2 + 3 = 3/4 - x, what is the value of x?
Solution :
x/2 + 3 = 3/4 - x
In the equation above, we find two fractions with two different denominators 2 and 4.
The least common multiple of (2, 4) = 4.
Multiply both sides of the above equation by 4 to get rid of the fractions.
4(x/2 + 3) = 4(3/4 - x)
4x/2 + 12 = 12/4 - 4x
2x + 12 = 3 - 4x
Add 4x to both sides.
6x + 12 = 3
Subtract 12 from both sides.
6x = -9
Divide both sides by 6.
x = -9/6
x = -3/2
Problem 6 :
If y - (3 - 2y) + (4 - 5y) = -7, what is the value of y?
Solution :
y - (3 - 2y) + (4 - 5y) = -7
Simplify the right side of the equation.
y - 3 + 2y + 4 - 5y = -7
1 - 2y = -7
Subtract 1 from both sides.
-2y = -6
Divide both sides by -2.
y = 3
Problem 7 :
Solve for y :
(4/5)(y - 5) - (1/5)(y - 10) = 22
Solution :
(4/5)(y - 5) - (1/5)(y - 10) = 22
Using Distributive property,
4y/5 - 4 - y/5 + 2 = 22
(4y - y)/5 - 2 = 22
3y/5 - 2 = 22
Add 2 to both sides.
3y/5 = 24
Multiply both sides by 5.
3y = 120
Divide both sides by 3.
y = 40
Problem 8 :
If three quarters of a number decreased by twenty is equal to eighty two, what is that number?
Solution :
Let n be the number.
(3/4)n - 20 = 82
3n/4 - 20 = 82
Add 20 to both sides.
3n/4 = 102
Multiply both sides by 4.
3n = 408
Divide both sides by 3.
n = 136
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