What are equivalent equations? Starting from an arbitrary equation, use the Properties of Equality to find the equivalent ones.
Original One-Step equation: x/2=4 Equivalent equations: x +4 = 12 and 3x=24.
Practice makes perfect
We are asked to write an arbitrary one-step equation and then find two equations that are equivalent to it. Let's use the one-step equation
x/2=4.
Remember that equivalent equations are those which have the same solutions. We can start by finding the solution of our original equation. For this, we will isolate x using inverse operations and the Properties of Equality.
Therefore, the solution to our equation is x = 8. Notice that the solution x = 8 is an equality. If we follow the Properties of Equality, we can modify the equation without altering it. Let's use the Addition Property of Equality to generate an equivalent equation starting from the solution of our original equation.
If & x = 8,
then &x + 4 = 8 + 4
and &x +4 = 12.
Hence, the equation x +4 =12 is an equivalent equation to the original. We can verify this by solving for x.
As we can see, the equation has the same solution, x=8. To find a second equivalent equation we proceed in a similar manner as before. Let's use the Multiplication Property of Equality this time.
if & x = 8,
then &x * 3 = 8 * 3
and & 3x = 24.
Let's now verify that 3x=24 is an equivalent equation to the original.
