Seeing Structure to Solve Equations


This lesson is about using the techniques we have learned in the previous lessons to solve variables in equations. We know our order of operations, the three properties (commutative, associative, and distributive), as well as how we can work with expressions and equations.

We will also introduce a new concept - that we can mathematically do something to one side of an equation, as long as we do it to the “other side” of the equation as well.

Properties of Equality

We have these Properties of Equality:

Think of saying 3 = 3, or, 3 is the same as 3. It is still true if we add 1, as long as it's to both sides of the equation. So, 3 + 1 = 3 + 1 or 4 = 4. As we can see, the equation is still true, as we have followed the correct steps. This process can be applied for addition, subtraction, multiplication, division, and many other mathematical techniques as long as it's done to both sides of the equation. We will use this technique to mindfully manipulate equations with variables in them, and to give us a value for that variable.

First, let's learn how to manipulate to solve for a variable.

The solutions in the above animation are repeated with a more detailed explanation below.

Question 1.1

We have the equation 3x + 4 = 16

List the operations (addition, multiplication... etc.) that have been applied to the variable x, on the left side of the equation, and list them with the correct order of operations.

This is our first step in learning how to manipulate equations to solve for our purposes.

The question asks us to list the operations we have done to the variable x on the left-hand side, which would be 3x + 4. Let’s imagine we would solve this as if we knew what x was.

So, for the left-hand side we have,

3x + 4

If we were to calculate this expression, knowing the value of x, the first thing we would do would be:

Times the x by 3,

The next step would be:

Add 4

As this is more theoretical, it is easy to forget our order of operations, so keep that in mind.


Solve this equation, 3x + 4 = 16, by “reversing” or “doing the opposite” of what has been done to the variable x, make sure we reverse on both sides of the equation, following the Additive Properties of Equality and the Multiplicative Properties of Equality. When you have calculated x, sub it back into the equation to double check it is correct.

The purpose of “reversing” or “doing the opposite” of what has been done to x is to get an equation that looks like x = …, this will give us the value for x.

So, remembering what has been done to x was:

Multiplied x by 3 then added 4

To be clear, adding is the opposite of subtracting, and multiplying is the opposite of division.

When we are rearranging to solve x, we do the opposite mathematical method and we do it in the opposite order and we do it to both sides of the equation. (Additive Properties of Equality and the Multiplicative Properties of Equality)

So, “reversing” 3x + 4 = 16 will look like:

Subtract 4 from both sides - this was our last step from above, so it is our first step when we reverse it.

Then, divide both sides by 3

So, First, we subtract 4 from both sides,

3x + 4 - 4 = 16 - 4

Gives us,

3x + 0 = 12

3x = 12

Now, we divide both sides by 3,

3x ÷ 3 = 12 ÷ 3

x = 4

Now we have our answer, x = 4, we sub this back into the original equation, if both sides are equal, we have a correct value for x.

Subbing x = 4 into 3x + 4 = 16

3(4) + 4 = 16

12 + 4 = 16

16 = 16

So, it is true. We can use this method to double check our answers anytime we are solving for a variable.

We just used everything we have learned so far to figure something we did not know! The world is ruled by the laws of math, and there are plenty of things we do not know, but we can use the mathematical methods and facts we do know, and find out things about the world we did not!

This method or “reversing” or “doing the opposite” using the Additive Properties of Equality and the Multiplicative Properties of Equality is called Inverse Operations. We can use this method only when we have an equation where the variable shows up once,

like 2x - 5 = 13 and not like x2 - 4 = 3x + 4 (do not worry about that, we will learn how to solve that later)


Question 1.2

Solve each equation for x by using Inverse Operations

i) identify what has been done to x and in what order

ii) Write the inverse operations and then apply them to solve for x, using the Additive Properties of Equality and the Multiplicative Properties of Equality


3(x + 1) - 4 = -10

i) Identify what has been done, and in what order:

Write BEDMAS to help remember, order of operations becomes a bit trickier when thinking about it in reverse,

We are just looking at the working with the side with the x on it, 3(x + 1) - 4

For now, we can first simply write the order of operations out in order, without the specific detail.

So, using BEDMAS we see the first think we would do to x is:

1) Brackets

Then the next is exponents, of which there are none, so next is multiply and/or divide. We have multiply, so,

2) Multiply

Lastly, we have the addition and subtraction. As we see we have subtraction, so,

3) Subtraction

Now we have broken this down a bit, lets add in the specific operations that are in each step.

In the brackets, we have x + 1, so this step is adding 1,

1) Add 1

Now we have the multiplication, multiplying by 3, so we write,

2) Multiply by 3

Lastly is the subtraction, subtracting by 4, so,

3) Subtract by 4

There we have identified the operations done to x, and the order that we do those operations in. If you are confident with this, you don't have to write out the order of operations (i.e. 1) brackets 2) multiply 3) subtract) and can just start at the stage we did just above. This was done to help you see the order of operations.

ii) Write the inverse operations and then apply them to solve for x

Now we apply the Inverse Operations. Start by writing the list above in the opposite order with the opposite operation.

So, write the Inverse Operations of, (color coded to help see the order reversed)

1) Add 1

2) Multiply by 3

3) Subtract 4

So, the Inverse Operations will be,

1) Add 4

2) Divide by 3

3) Subtract 1

Now, to solve for x, we apply these steps of Inverse Operations to the full equation.

3(x + 1) - 4 = -10

1) Add 4 (to both sides! Additive Properties of Equality and the Multiplicative Properties of Equality)

3(x + 1) - 4 + 4 = -10 + 4

3(x + 1) + 0 = -6

3(x + 1) = -6

2) Divide by 3

3(x + 1)
3
  =  
-6
3

(x + 1) = -2

3) Subtract 1,

x + 1 - 1 = -2 - 1

x + 0 = -3

x = -3


x - 2
5
  + 12 = 6

i) identify what has been done to x and in what order

So first, we write what happened to x, on the left-hand side, following our order of operations (BEDMAS)

Again, we will first just write what the order of operations are, not with the specific operations. Then we fill in the specific operation, once we have written out the order. If you like, you can skip to writing both the order and the operations in the same step.

(x - 2)
5
  + 12

1) We have brackets, (remember, there are implied brackets for the numerator in division, it helps to write brackets around the numerator anytime you see something like:

x - 2
5

2) Division

3) addition

So, filling in the specific operations,

In the brackets we have x - 2 so,

1) Subtract 2

2) Divide by 5

3) Add 12


ii) Write the inverse operations and then apply them to solve for x

So, writing the Inverse Operations we have,

1) Subtract 12

2) Multiply by 5

3) Add 2

Now apply the Inverse Operations to both sides of the equation

1) Subtract 12 (from both sides!)

x - 2
5
  + 12 = 6
x - 2
5
  + 12 - 12 = 6 - 12
x - 2
5
  = -6

2) Multiply by 5, (when multiplying or dividing each side of an equation, it may help to write brackets around each side of the equation, then multiply or divide)

x - 2
5
 x 5 = (-6) x 5

x - 2 = - 30

3) Add 2

x - 2 +2 = -30 +2

x = - 28

There are multiple ways we can solve equations to give us answers to variables. The Inverse Operations is one of them, and generally, is the underlying way of solving equations. We can also apply other methods first before we apply the Inverse Operations. We do this as there can be ways to simplify an equation to make it easier to solve. Below we will use the original Inverse Operations method and, the Distributive Property & the Inverse Operations to the same equation.

The solutions in the above animation are repeated with a more detailed explanation below.

Question 2

Nine less than three times a number makes twenty-seven

i) Write the above statement in algebraic language

First, we are translating the language of English into the language of algebra (we are fluent in both!). Start by looking at the sentence, thinking about what each word or combination of words means and think if there is an algebraic symbol for it.

Nine less than three times a number makes twenty-seven.

Immediately, lets change the words of numbers into symbols of numbers.

9 less than 3 times a number makes 27.

We should recognize “less than” as subtraction, and “times” as multiply, and “makes” as equals, when it says “a number” that means it's the part we don't know, so, it's our unknown variable, x,

3 (a number ) - 9 = 27

3x - 9 = 27,

It is important to see that “9 less than” means something minus 9 and not 9 minus something,


Nine less than three times a number makes twenty-seven

ii) Solve the equation, you can use x to represent the unknown variable

Using Inverse operations on our equation,

3x - 9 = 27,

We will now skip the part writing the order of operations done to x. If you are struggling, go back and look at previous examples.

3x - 9 = 27

Doing the opposite of what has been done to x on both sides

Add 9

3x - 9 + 9 = 27 + 9

3x = 36

Divide both sides by 3

(3x) ÷ 3 = (36) ÷ 3

x = 12


Nine less than three times a number makes twenty-seven

iii) Check your answer is correct by subbing the unknown variable back into your equation.

So, sub x = 12 into, 3x - 9 = 27, if the equation is true, we have the right value for x,

3(12) - 9 = 27

36 - 9 = 27

27 = 27

Therefore, x = 12 is correct.

Recap

In this lesson we learned how to solve equations for a variable We did this by:

  1. Identifying the order of operations - making a list of the order of operations that you would do to the variable
  2. Using the inverse order of operations and reversing each operation itself - reverse the order of operations, and do the inverse operation in each step (i.e. subtract 2 becomes add 2, divide by 3 becomes multiply by 3)
  3. Applying the Inverse operations and solve for the variable - Follow all the correct steps and you will be left with your variable isolated giving you and answer.