Equivalent Equations – Definition and Examples

                                                                                         

The term “equivalent” means “the same”. Two equations are said to be equivalent if they have the same solution.

Let’s look at a few examples.

• y = 3 and 2y – 6 = 0 are equivalent equations.
y = 3 is a simple linear equation. The other equation 2y – 6 = 0 can be simplified to y = 3 upon dividing by 2.

• The equations 2x = x and 2x + 3 = x + 3 are equivalent, because, they have the same solution x = 0.

• x^3 – 64 = 0 and x = 4 are equivalent equations.
x = 4 is a simple linear equation. The cubic equation x^3 – 64 = 0 has the solution x = 4. When we replace x with 4, we get: 4^3 – 64 = 64 – 64 = 0.

• x^2 – 5x + 6 = 0 and (x – 3)(x – 2) = 0 are equivalent equations.
x^2 – 5x + 6 = 0 is a quadratic equation. (x – 3)(x – 2) = 0 is the factored form of the same equation.

• Logarithms and Exponents are inverse functions. Log (base 2) x = 3 and 2^3 = x are equivalent equations, because they have the same solution x = 8.

More about Equivalent Equations

We can create equivalent equations using the following methods:

(a) Add or subtract the same quantity on either sides of an equation.
(b) Multiply or divide both sides of an equation by the same quantity.

Let’s create equations equivalent to 5a = 20 using the methods mentioned above.

• Add 3 to each side.
5a + 3 = 20 + 3 gives 5a + 3 = 23

• Subtract 2 from each side.
5a – 2 = 20 – 2 produce an equivalent equation 5a – 2 = 18

• Multiply each side by 4.
5a × 4 = 20 × 4 gives 20a = 80

• Divide each side by 5.

5a ÷ 5 = 20 ÷ 5 gives rise to an equivalent equation a = 4

Solved Example on Equivalent Equations
Which of the following equations is equivalent to the equation 3(x – 5) = 4x?
A. 3x = – 45
B. 3x = 45
C. 2x = 15 – x
D. x + 3 = – 18

Solution:

Step 1: Let us first solve the given equation 3(x – 5) = 4x.
Step 2: Distribute the 3 that is outside the parenthesis to every term inside the parenthesis and multiply. We get:

3x – 15 = 4x

Step 3: 3x – 4x – 15 = 0 [Combine the terms containing ‘x’.]
Step 4: – x – 15 = 0 [Simplify.]
Step 5: – x – 15 + 15 = 15 [Add 15 to each side.]
Step 6: – x = 15 is the same as x = – 15 [Simplify.]

Step 7: Now, we need to identify from the choices listed the equation with the same solution (–15).
Step 8: Among the given choices, only 3x = – 45 simplifies to x = – 15 upon dividing both sides by 3.
Step 9: So, 3x = – 45 is the equation equivalent to the given equation.


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