1.2. Equations
Equations
Equation: a mathematical sentence that contains two expressions and is separated by an equal sign (both sides of the equation have the same value).
To solve an equation we find a particular value for the variable in the equation that makes the equation true (left side = right side).
Example: For the equation x + 4 = 5
only x = 1 can make it true, since 1 + 4 = 5 (Left side = Right side)
Solution of an equation: the value of the variable in the equation that makes the equation true.
Example 1.2.1
Indicate whether each of the given number is a solution to the given equation.
| 1) 2 : 4x – 3 = 5 | 4 ∙ 2 – 3 5 | 5 5 | Yes | Replace x with 2. |
| 2) 15 : y = -3 | -3 | -3 -3 | Yes | Replace y with 15. |
| 3) : 8t = 3 | 8 () 3 | 4 ≠ 3 | No | Cannot replace t with . |
Solving Equations
Basic rules for solving one-step equations:
- Add, subtract, multiply or divide the same quantity to both sides of an equation to obtain a valid equation.
- Remember to always do the same thing to both sides of the equation (balance).
Properties for solving equations:
| Properties | Equality | Example |
| Addition property of equality | A = B A + C = B + C | Solve
x = 9 |
| Subtraction property of equality | A = B A – C = B – C | Solve
y = -13 |
| Multiplication property of equality | A = B A · C = B · C | Solve
m = 18 |
| Division property of equality | A = B (C ≠ 0) | Solve
n = –5 |
Example 1.2.2
Solve the following equations.
| 1) | Property of addition. | |
| Check: | Replace x with 14. |
| 2) | Property of subtraction. | |
| 3) | Property of multiplication. | |
| 4) | Property of division. | |
Multi-step equation: an equation that requires more than one step to solve it.
Procedure for solving multi-step equations:
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| Steps | Example |
| Solve | ||
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Multiply each term by 5. | |
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Subtract 10 from both sides.
Subtract 6y from both sides. |
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Divide both sides by -5. | |
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Replace y with 2.
Multiply each term by 5.
LS = RS (correct) |
Equations involving decimals: Multiply every term of both sides of the equation by a multiple of 10 (10, 100, 1000, etc.) to clear the decimals (based on the number with the largest number of decimal places in the equation).
| Steps | Example |
| Solve | ||
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The largest number of decimal place is two. | |
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Add 12 to both sides. Add 426x to both sides. |
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Example 1.2.3
| Solve 0.4y + 0.08 = 0.016 | The largest number of decimal place is three. |
| 1000(0.4y) + 1000(0.08) = 1000(0.016) | Multiply each term by 1000. |
| 400y + 80 = 16 | Combine like terms. |
| 400y = -64 | Divide both sides by 400. |
| y = – 0.16 |
Equations involving fractions:
| Steps | Example |
| Solve | ||
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Add 6t to both sides. Subtract 9 from both sides. |
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Divide both sides by 10. |
Word Problems
Identifying keywords:
- When trying to figure out the correct operation ( + , – , × , ÷ , etc.) in a word problem it is important to pay attention to keywords (clues to what the problem is asking).
- Identifying keywords an