71 lines
2.3 KiB
Markdown
71 lines
2.3 KiB
Markdown
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---
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tags:
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- Mathematics
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- Algebra
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---
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## Equivalent equations
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>
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> Two equations are equivalent if they have the same [solution](Algebra%20key%20terms.md#678811) set.
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We know from the distributive property of multiplication that the equation $a \cdot (b + c )$ is equivalent to $a \cdot b + a \cdot c$. If we assign values to the variables such that $b$ is $5$ and $c$ is $2$ we can demonstrate the equivalence that obtains in the case of the distributive property by showing that both $a \cdot (b + c )$ and $a \cdot b + a \cdot c$ 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 (the assertion that arrangement of values results in $14$). Since both expressions have the same solution they are equivalent.
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## Creating equivalent equations
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We can create equivalent equations by adding, subtracting, multiplying and dividing the *same quantity* from both sides of the equation (i.e. either side of the $=$ symbol).
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Adding or subtracting the same quantity from both sides (either side of the $=$ ) of the equation results in an equivalent equation.
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### Demonstration with addition
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$$ x - 4 = 3 $$
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The [solution](Algebra%20key%20terms.md#678811) to this equation is $7$
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$$ x - 4 (+4) = 3 (+ 4) $$
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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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$$ x + 4 = 9 $$
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The [solution](Algebra%20key%20terms.md#678811) to this equation is $5$.
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$$ x + 4 (-4) = 9(-4) $$
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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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### Demonstration with multiplication
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$$x \cdot 2 = 10 $$
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The [solution](Algebra%20key%20terms.md#678811) to this equation is $5$.
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$$ (x \cdot 2) \cdot 3 = 10 \cdot 3 $$
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Here we have multiplied each side of the equation by $3$. If $x =5$ then
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$$ (5 \cdot 2) \cdot 3 = 30$$
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$$ 10 \cdot 3 = 30$$
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### Demonstration with division
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$$x \cdot 3 = 18 $$
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The [solution](Algebra%20key%20terms.md#678811) to this equation is $6$.
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$$\frac{x \cdot 3}{3} = \frac{18}{3} $$
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Here we have divided each side of the equation by $3$. If $x$ is 6, then
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$$\frac{6 \cdot 3}{3} = 6$$
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$$\frac{18}{3} = 6 $$
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