Methods of Solving Equations
An equation has a letter, an ‘equals’ sign and has two sides, as in this example:
To solve an equation we find the value of the letter.
Method
Remove all the terms from one side of the equation, leaving the letter by itself.
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y + 5 = 7 |
(– 5 from both sides) |
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To remove the + 5 we subtract 5. To keep the two sides equal, we must take 5 from both sides.
So
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y = 7 – 5
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y = 2 is the solution to the equation. |
Example 1: Solve y + 8 =
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11. |
(– 8 from both sides)
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y = |
11 – 8 |
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y = |
3 |
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Example 2: Solve y + 2 =
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7. |
(– 2 from both sides)
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y = |
7 – 2 |
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y = |
5 |
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Note: a quick way of solving this equation is to move the + 2 to the other side and change its sign to – 2.
Example 3:
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y + 10 = 14
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(– 10 from both sides)
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y = 14 – 10
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y = 4 |
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Example 4:
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y – 6 = 2
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(+ 6 to both sides)
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y = 2 + 6
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Note: we add 6 in this case |
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y = 8 |
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Example 5:
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3y = 15
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(÷ both sides by 3)
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y = 15 ÷ 3
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We change the multiply by 3 to divide by 3.
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y = 5 |
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Example 6:
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y = 2
7
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(x both sides by 7) |
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y = 2 x 7
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We change the divide by 7 to multiply by 7. |
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y = 14 |
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General Rule
An important general rule is that if we move a term to the other side of the equation we change its sign.
If there is more than one term to move, always move the + or – term first.
| Example 1: Solve 2x + 5 = |
15. |
(– 5 from both sides) |
2x = |
15 – 5 |
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2x = |
10 |
(then ÷ 2)
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x = |
10 ÷ 2 |
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x = |
5 |
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| Example 2: Solve |
x – |
4 = 2. |
(+ 4 to both sides) |
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3 |
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x = 2 + 4 |
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3 |
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x = 6 |
(x both sides by 3) |
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3
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x = 6 x 3
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x = 18 |
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Brackets
Brackets are used to group terms together.
If we want to remove brackets, then everything inside the bracket must be multiplied by the term on the outside.
3 (y + 2) = 3 x y + 3 x 2 = 3y + 6
Both y and the + 2 must be multiplied by 3.
5 (y – 3) = 5 x y – 5 x 3
= 5y – 15
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Equations with Brackets
Example 1: Solve 5 (y – 3) = 20.
Remove the brackets.
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5y – 15 = 20 |
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5y = 20 + 15 |
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5y = 35 |
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y = 35 ÷ 5 |
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y = 7 |
| Example 2: Solve |
p + 4 = 5. |
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3 |
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Note: This line brackets p + 4 together. So we cannot (– 4) first.
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p + 4 = 5 x 3
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p + 4 = 15
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(– 4 from both sides) |
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p = 15 – 4
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p = 11 |
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Equations with Letters on Both Sides
Example 1: Solve
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3 (y –1) = y + 7.
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3y – 3 = y + 7
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(+ 3 to both sides)
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3y = y + 10
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(– y from both sides)
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The y terms must be brought to the same side and simplified. |
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3y – y = 10
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2y = 10
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(÷ 2)
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y = 5 |
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Example 2: Solve
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4 (p + 2) = 18 – p.
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(add p to both sides)
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5p + 8 = 18
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(– 8 from both sides)
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5p = 10
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(÷ 5)
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p = 2 |
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