Equações com produtos equivalentes

"Producing equivalent equations" sounds daunting, but we do it all the time! Equations are two chunks of math (expressions) set equal to each other. $$4x + 3 = 2x - 2$$

We manipulate both sides of equations to produce equivalent equations. These have the same relationship, but give us more useful information. We typically use it to solve for a variable, such as solving for the value of $x$.

So how do we do it?

Just by taking the steps you would normally take to solve an equation! Performing an operation on both sides, swapping the sides, or combining like terms will all produce equivalent equations. You can also factor within a single side.

Think of an equation like two sides of a scale. If they're at equal weight, you have to do the exact same thing on both sides to keep the weights the same. Combining like terms and factoring is just moving stuff on one side around. They don't actually changing anything.

On the other hand, if we DIDN'T do the same thing on both sides, the scale would go lopsided:

Whenever you are playing with equations, remember: whatever you do to one side of the equation, you have to do on the other side. Keeping this in mind will help you to make sure all of your future equations remain equivalent.

Equivalent equations don't always look similar to one another. Manipulating both sides with the same operation on either of them can make them mathematically identical.

(Video) Equivalent Equations

by TheMrAyre

This video by TheMrAyre discusses how to keep two sides of an equation equal.

$\large{\text{Summary}}$

You can keep the two sides of an equation equal by:

Adding a number to both sides of the equation $$\begin{align} 5a-3&=12\\ 5a -3 \,{\color{#8925AE}{+\,3}} &= 12\, {\color{#8925AE}{+\,3}}\\ 5a&= 15\end{align}$$

Subtracting a number from both sides of the equation $$\begin{align}10+2x&=36\\ 10+2x \,{\color{#8925AE}{-\,10}} &= 36\, {\color{#8925AE}{-\,10}}\\ 2x &= 26 \end{align}$$

Multiplying both sides of the equation by a number $$\begin{align}\frac{x}{6} &= 3\\ \frac{x}{6}\, {\color{#8925AE}{ \times \,6}} &= 3\, {\color{#8925AE}{ \times \,6}}\\ x &= 18 \end{align}$$

Dividing both sides of the equation by a number $$\begin{align} 8y&=40\\ \frac{8y}{\color{#8925AE}{8}} &= \frac{40}{\color{#8925AE}{8}}\\ y&= 5\end{align}$$

To solve equations (in other words, find out what values two expressions are equal for), we need to use these operations to keep transforming equations into simpler and simpler ones. Our goal is to get the variable on one side of the equation and find its value. It gets a little trickier when there are multiple variables, but these rules stay the same!

Equivalent Equations

Two equations are equivalent if you can manipulate one equation to get the other equation. We must be able to start at either equation and end up at the other.

So, the question is, how does each possible step work? There are four different strategies that you can use to make equivalent equations: addition, subtraction, multiplication, and division.

We'll go through each type of step you can use with specific examples. Then we'll do a longer example at the end that combines multiple steps.

Strategy 1: Addition

We can add any number to both sides of an equation, but the number we add to both sides must be the same. If not, then we no longer have an equation, and any algebra we do after that is pointless.

Example 1: If we have the equation $$x - 4 = 2x + 7,$$ we can add $4$ to both sides of the equation to get an equivalent equation: $$x - 4 + 4 = 2x + 7 + 4.$$

Of course, we can simplify both sides of the equation. The constants on the left hand side cancel out: $$x = 2x + 7 + 4.$$

We can also add $7 + 4$ together on the right side: $$x = 2x + 11.$$

Let's move on to our next strategy, the reverse of addition.

Strategy 2: Subtraction

The reverse of addition is subtraction. This means if we want to get rid of a $+$ something on one side of an equation, we should subtract that something from both sides of the equation.

Example 2: Let's say we have the equation $$4 + 2x = 6x.$$

One option is to subtract $2x$ to both sides. The reason we'd want to do this is because there is a $+2x$ on the left side, so subtracting $2x$ gets rid of the $+$ term. $$4 + 2x - 2x = 6x - 2x.$$

As said before, the $-2x$ and the $2x$ on the left side cancel out: $$4 = 6x - 2x.$$

We can simplify the right-hand side since both terms are $x$-terms: $$4 = 4x.$$

Addition and subtraction are a pair of operations we can do. The other pair is multiplication and division.

Strategy 3: Division

If we have a factor in front of a term we want to get rid of, we can divide it away. We will NEVER divide by $0$ because we just aren't allowed to.

Example 3: Let's say we have this equation: $$12x = 24.$$

Many times, the goal in an equation is to get the $x$ (or other variable) by itself. We can do this by dividing both sides of the equation by $12:$ $$\frac{12x}{12} = \frac{24}{12}.$$

We choose to divide by $12$ because $x$ was being multiplied by $12,$ and these two operations cancel out. So: $$x = \frac{24}{12}$$ $$x=12.$$

Strategy 4: Multiplication

At last, the last operation is multiplication, or as I like to call it, "backwards division." We can get rid of division by multiplying both sides of the equation by a chosen number. We will not multiply by $0$ because the equation that results will be $0 = 0,$ which is not helpful in any case. $0 = 0$ is always true even if the equation we started with had a variable in it.

Example 4: Maybe we have the equation $$\frac{17x + 4}{2} = 5.$$

Equations generally look nicer and are easier to handle if they don't have fractions. How do we handle this? Get rid of the fraction! The fraction involves a division by $2,$ so we do the opposite. We multiply both sides of the equation by $2$ to undo this: $$2 \times \frac{17x + 4}{2} = 2 \times 5.$$

The multiplication and division on the left-hand side cancel: $$17x + 4 = 10.$$

We haven't fully solved the equation for $x,$ but the point was to show how multiplication works.

Now we'll do a longer example where we work in both directions. We show both directions because the steps must be reversible to have two equations that are equivalent.

Combining the 4 Strategies

Our example is to show that the following two equations are equivalent: $$\frac{x}{6} + 3 = 5$$ and $$2x - 17 = 7.$$

We assume that $\frac{x}{6} + 3 = 5$ and use operations to show that $2x - 17 = 7.$ Then we go backwards. Think of this as combining the tricks of Examples 1, 2, 3, and 4. Watch out for the fraction!

We start with $$\frac{x}{6} + 3 = 5,$$ but how do we know which strategy to use? There are a bunch of choices that will actually work, so we'll pick one. Our starting equation has $\frac{x}{6}$ and our "finished" equation has a $2x$, so what number could we multiply both sides of the equation by to turn $\frac{x}{6}$ into $2x$?

We'll multiply both sides of the equation by $12:$ $$12 \times \frac{x}{6} + 12 \times 3 = 12 \times 5.$$

The first term on the left-hand side is a $2x:$ $$2x + 12 \times 3 = 12 \times 5.$$

The other multiplications are just between numbers: $12 \times 3 = 36$ and $12 \times 5 = 60,$ so $$2x + 36 = 60.$$

Now what? We have a $2x$ like we wanted, but the constant terms need some work. The "finished" equation has a $-17$ on the left side, so we want to make the constant term on the left smaller.

Subtract $53$ from both sides of the equation: $$2x + 36 - 53 = 60 - 53.$$

Now simplify: $$2x - 17 = 7.$$

Tada! We have the equation we wanted. But this is only half of the battle.

Now we have to go backwards. We assume that $$2x - 17 = 7.$$

This time we want to use addition, subtraction, multiplication, and division to show that $$\frac{x}{6} + 3 = 5.$$

However, this time our job is much easier. In the first part of the example we picked steps that could be reversed. Now we just list out those steps in reverse. Our last step in the first part was subtracting $53$ from both sides of the equation, so now we do the opposite: add $53$ to both sides: $$2x - 17 + 53 = 7 + 53.$$

Simplifying by adding terms together gives $$2x + 36 = 60.$$

This was one of our "in-between" steps earlier, which is always a good sign that we're doing things correctly. We also multiplied both sides of the equation by $12$ in the first part of the example, so once again we do the reverse: divide by $12:$ $$\frac{2x + 36}{12} = \frac{60}{12}.$$

Lastly, we can simplify this as $$\frac{x}{6} + 3 = 5.$$

Now we're done.

Summary

Two equations are equivalent if we can add, subtract, multiply, and divide numbers from one equation to get the other, regardless of which equation we start with. In doing this it is helpful to remember that addition and subtraction are reverse operations, as are multiplication and division.