7.04 Methods for solving two-step equations

When working with expressions that involve a pronumeral, we can see how they were constructed by observing the order in which operations were applied to that pronumeral.

We have seen this kind of thinking with simple equations already. For example, the expression $x+4$x+4 can be broken down into 'an unknown number plus four', where the unknown number is $x$x and the operation is '$+4$+4'.

In this lesson, we will look at how we can apply this concept to more complicated expressions and then use that to solve equations with those expressions.

Buildng expressions

Before we start breaking expressions down, we should first understand how they are built up.

Consider the operations 'multiply by $6$6' and 'add $9$9'.

If we apply these operations to some pronumeral, we can build an expression:

Do we get the same expression if we apply the operations in a different order?

No.

Although these two expressions are built from the same operations, they are different because the operations were applied in a different order.

Notice that the 'multiply by $6$6' operation was only applied to $y$y in the first expression, while it was applied to $y+9$y+9 in the second.

Practice question

Question 1

The following operations are performed on $s$s.

	$-$−$7$7		$\times$×$6$6
$s$s		$\editable{}$		$\editable{}$

1. The value in the first blank will be:

   $7-s$7−s

   A

   $-7s$−7s

   B

   $s-7$s−7

   C

   $s+7$s+7

   D

2. The value in the 2nd blank will be:

   $\frac{s-7}{6}$s−76​

   A

   $6+s-7$6+s−7

   B

   $6\left(s-7\right)$6(s−7)

   C

   $6s-7$6s−7

   D

Reveal SolutionWatch video

Breaking down expressions

When breaking down expressions, our aim is to apply operations that will turn the expression into an isolated pronumeral.

Returning to the expression $6\left(y+9\right)$6(y+9), we can see that applying the operations 'divide by $6$6' and 'subtract $9$9' will turn the expression back into an isolated pronumeral:

Notice that when we built the expression $6\left(y+9\right)$6(y+9), we applied the operations:

1. Add $9$9
2. Multiply by $6$6

And when we broke the expression back down to $y$y, we applied the operations:

1. Divide by $6$6
2. Subtract $9$9

We can see that, when breaking down an expression, we can reverse the operations used to build the equation (and the order in which they are applied) to cancel out the operations applied to the pronumeral.

In the images below, we can see that the operation 'divide by $6$6' cancels out the multiplication in the expression, then 'subtract $9$9' cancels out the addition.

The 'divide by $6$6' and 'multiply by $6$6' operations cancel each other out.

The 'subtract $9$9' and 'add $9$9' operations cancel each other out.

Notice again that the order in which we apply the operations is important. We can see in the image that the operations that wil cancel out must come one after the other. If this is not the case, we will get something like this:

When operations do not cancel or simplify, the expression becomes very complicated.

If we apply our reverse operations in the wrong order, our expression will get even more complicated!

Solving equations

Now that we have a way to isolate pronumerals in an expression, we can apply this method to solve equations that involve more than one step. We can do this since, as long as we apply an operation to both sides of the equation, expressions on either side of the equals sign will be equal in value.