52 lines
1.3 KiB
Markdown
52 lines
1.3 KiB
Markdown
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---
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tags: [algebra, exponents]
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---
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## Equivalent equations
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> Two equations are equivalent if they have the same solution set.
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We know from the distributive property of multiplication that the equation
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$a \cdot (b + c )$ is equivalent to $a \cdot b + a \cdot c$. If we assign values
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to the variables such that $b$ is equal to $5$ and $c$ is equal to $2$ we can
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demonstrate the equivalence that obtains in the case of the distributive
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property by showing that both $a \cdot (b + c )$ and $a \cdot b + a \cdot c$
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have the same solution:
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$$ 2 \cdot (5 + 2) = 14 $$
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$$ 2 \cdot 5 + 2 \cdot 2 =14 $$
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When we substitute $a$ with $2$ (the solution) we arrive at a true statement
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(the assertion that arrangement of values results in $14$). Since both
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expressions have the same solution they are equivalent.
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## Creating equivalent equations
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Adding or subtracting the same quantity from both sides (either side of the $=$
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) of the equation results in an equivalent equation.
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### Demonstration with addition
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$$ x - 4 = 3 \\ x -4 (+ 4) = 3 (+ 4) $$
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Here we have added $4$ to each side of the equation. If $x = 7$ then:
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$$ 7 - 4 (+ 4) = 7 $$
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and:
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$$ 3 + 4 = 7 $$
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### Demonstration with subtraction
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$$ x + 4 = 9 \\ x + 4 (-4) = 9 (-4) $$
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Here we have subtracted $4$ from each side of the equation. If $x = 5$ then:
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$$ 5 + 4 (-4) = 5 $$
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and
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$$ 9 - 4 = 5 $$
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