Equivalent equations

Two equations are equivalent if they have the same solution set.

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:

 2 \cdot (5 + 2) = 14

 2 \cdot 5 + 2 \cdot 2 =14

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.

Creating equivalent equations

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). Adding or subtracting the same quantity from both sides (either side of the = ) of the equation results in an equivalent equation.

Demonstration with addition

x - 4 = 3

The solution to this equation is 7

x - 4 (+4) = 3 (+ 4)

Here we have added 4 to each side of the equation. If x = 7 then:

 7 - 4 (+ 4) = 7

and:

 3 + 4 = 7

Demonstration with subtraction

x + 4 = 9

The solution to this equation is 5.

x + 4 (-4) = 9(-4)

Here we have subtracted 4 from each side of the equation. If x = 5 then:

 5 + 4 (-4) = 5

and

 9 - 4 = 5

Demonstration with multiplication

x \cdot 2 = 10 The solution to this equation is 5.

(x \cdot 2) \cdot 3 = 10 \cdot 3 Here we have multiplied each side of the equation by 3. If x =5 then

 (5 \cdot 2) \cdot 3 = 30

 10 \cdot 3 = 30

Demonstration with division

x \cdot 3 = 18

The solution to this equation is 6. $$\frac{x \cdot 3}{3} = \frac{18}{3}

Here we have divided each side of the equation by 3. If x is 6, then

\frac{6 \cdot 3}{3} = 6

\frac{18}{3} = 6

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