High School Algebra I Unlocked (2016)

Lesson 4.3. Equivalent Equations

Recall that, unlike an expression, an equation contains an equals sign. All of the following are equations:

2x + 6 = 12 + x   2x = 6 + x   x = 6

Not only are all of these equations, but they are also equivalent equations, or equations that have the exact same solution set. Let’s take a closer look at the first equation in the set above and solve for x.

First, subtract 6 from both sides of the equation to find that

2x + 6 = 12 + x

2x + 6 − 6 = 12 + x − 6

 2x = 6 + x

Notice that this equation is equivalent to the second equation in our original set. Now subtract x from both sides of the equation.

2x = 6 + x

2x − x = 6 + x − x

x = 6 

As you can see, the resulting equation, x = 6, is identical to the final equation in the original set above, 2x + 6 = 12 + x, 2x = 6 + x, and x = 6.

While all of the equations listed above appear to be different, they are all equivalent, providing the same value for x when solved.

Start by combining like terms, subtracting 6s from both sides of the inequality to find that 7s − 14 < 4 + 6s is equivalent to s − 14 < 4. Next, add 14 to both sides of the inequality to find that s − 14 < 4 becomes s < 18. Therefore, the correct answer is (D).

Let’s try a question that tests your ability to identify equivalent equations.

EXAMPLE

Which of the following equations is equivalent to 18x + 14 = 22 + 14x ?

A) 8x = 4

B) 4x = 8

C) x =

D) x = −2

In order to find the equation that’s equivalent to 18x + 14 = 22 + 14x, you need to work the equation and solve for x. Start by subtracting 14x from both sides of the equation to find that

18x + 14 = 22 + 14x

18x + 14 − 14x = 22 + 14x − 14x

4x + 14 = 22

Next, subtract 14 from both sides of the equation.

4x + 14 = 22

4x + 14 − 14 = 22 − 14

 4x = 8

Before you continue, take a look at the answer choices. Do you see a choice that matches? If you said (B), you’re correct! You don’t even need to solve for x to find the equivalent equation.

But look at what happens when the question is slightly different.

EXAMPLE

Which of the following equations is equivalent to 18x + 14 = 22 + 14x ?

A) 8x + 2 = 4x − 8

B) −4x − 6 = 8x − 14

C) 2x = x + 4

D) x + 9 = −2x + 15

Just as in the previous question, you need to find an equation that is equivalent to 18x + 14 = 22 + 14x. But in this scenario, you have to work the entire equation and solve for x.

Again, start by subtracting 14x from both sides of the equation to find that

18x + 14 = 22 + 14x

18x + 14 − 14x = 22 + 14x − 14x

4x + 14 = 22 

Next, subtract 14 from both sides of the equation.

4x + 14 = 22

4x + 14 − 14 = 22 − 14

 4x = 8

Then divide both sides of the equation by 4 to solve for x.

4x = 8

x = 2

Unlike Example 9, the answer doesn’t jump out at you here; there is no answer choice that states x = 2. So now you have two options: You can solve each equation in the answer choices and find the one that provides the solution x = 2, or you can substitutex = 2 into each answer choice to see if the equation holds true.

If you choose to solve each equation in the answer choices, you will follow the same process that you did to solve the equation in the question. Thus, you would solve (A) to find that:

8x + 2 = 4x − 8

8x − 4x = −8 − 2

4x = −10

 x = −

Since this is a false statement, and x ≠ −10/4, (A) can be eliminated. Repeat this process with (B) to find that

−4x − 6 = 8x − 14

−4x − 8x = −14 + 6

−12x = −8

12x = 8

x =

Similarly, this is a false statement, and x ≠ 8/12, so (B) can be eliminated. Repeat this process again for (C):

2x = x + 4

2x − x = 4  

x = 4 

Once again, this is a false statement, as x ≠ 4, so (C) can be eliminated. Finally, solve for (D):

 x + 9 = −2x + 15

x + 2x = 15 − 9

3x = 6

x = 2

Here, x = 2 is true, so the equation in (D) is equivalent to the equation given in the question.