Explainer: One-step Equations: Multiplication and Division

In this explainer, In this explainer, we will learn how to write and solve one-step multiplication and division equations in questions including word problems.

First, make sure you are familiar with the key vocabulary we will be using.

Definition: Key Vocabulary

  • Linear Equations: They are equations that give a straight line when plotted on a graph; for example, ๐‘ฅ+2=5,๐‘ฅโˆ’3=7,2๐‘ฅ=10,๐‘ฅ3=4.and
  • One-Step Equations: They are equations that only require one calculation to solve.
  • Reciprocal Operations/Inverse Operations: These are operations that can undo a calculation. The reciprocal operation of adding is subtracting, the reciprocal operation of multiplying is dividing, and vice versa.
  • Solving an Equation: It means Finding the value of ๐‘ฅ that makes the equation true.

When solving equations, we have to remember to perform the same operation on both sides of the equal sign; otherwise, the equation will no longer be balanced. It can help to think of the equation as being on a scale. We will start with an example of solving a one-step equation by thinking about it in this way.

Example 1: Solving One-Step Linear Equations by Balancing Both Sides

Solve ๐‘ฅโˆ’3=10.

Answer

It helps to think of the equation as being on a pair of scales. The two sides of the equation are equal and so they balance. Any change we make to one side we also have to make to the other side to keep the scales balanced and the equation true.

To solve the equation, we want to rearrange it to get ๐‘ฅ on its own on one side of the balance. To do this, we need to use a reciprocal operation to get rid of โˆ’3 from the left side.

If we add 3 to the left-hand side, then we would have ๐‘ฅโˆ’3+3 on that side which is equal to ๐‘ฅ+0. So, the +3 cancels with the โˆ’3 to give us ๐‘ฅ, which is what we want. But if we just did this, the scales would no longer balance. So, we add 3 to both sides.

Now that the scale is balanced, we can cancel and simplify to find the solution.

By adding 3 to both sides, we have found that the solution is ๐‘ฅ=13.

We can check that this makes sense by substituting this value of ๐‘ฅ into the equation. Indeed, 13โˆ’3=10 and this verifies that ๐‘ฅ=13 is the solution.

It is not necessary to draw the balance each time; just use the idea to help you remember to always perform the same operation on both sides. Now we will look at another example which will show you what you need to write when answering these questions.

Example 2: Solving One-Step Linear Equations Involving Addition and Subtraction

If ๐‘ฅ+2=5, then what is the value of ๐‘ฅ?

Answer

We start by writing what we know, that ๐‘ฅ+2=5.

Then, we need to figure out how we can use reciprocal operations to get ๐‘ฅ on its own on the left-hand side. This means that we need to cancel +2. Since the inverse of adding is subtracting, we will subtract 2 from both sides, so ๐‘ฅ+2โˆ’2=5โˆ’2.

Now, on the left-hand side, +2 and โˆ’2 cancel to give 0, since 2โˆ’2=0, and the right-hand side simplifies to 3. We have shown that ๐‘ฅ=3.

Further Comments

When writing a solution, you would normally be expected to write these three equations to show each step. It is also useful to make a note of the reciprocal operations you used each time.

A good answer is shown below; in the second line, we have indicated that we subtracted 2 from both sides and crossed out the terms that cancel.

Now that we have seen how to use reciprocal operations to keep our equation balanced and how to record our answers, we will look at a few more examples.

Example 3: Solving One-Step Linear Equations Involving Addition and Subtraction

Solve ๐‘ฅ+7=7.

Answer

To get rid of +7 on the left-hand side, we need to subtract 7 from both sides: ๐‘ฅ+7=7๐‘ฅ+7โˆ’7=7โˆ’77๐‘ฅ=0+7โˆ’7.(subtractfrombothsides)(canceland)

So, the solution is ๐‘ฅ=0.

Further Comments

Notice that, in this example, the numbers on the right-hand side canceled as well. There is nothing wrong with getting an answer of ๐‘ฅ=0.

It is good practice to check your answer by substituting it back into the original equation. When ๐‘ฅ=0, we get that ๐‘ฅ+7=0+7=7 which is what we expected.

Example 4: Solving One-Step Linear Equations Involving Addition and Subtraction

Solve ๐‘ฅ+8=5.

Answer

To get rid of +8 on the left-hand side, we need to use the reciprocal operation of addition, which is subtraction. Subtracting 8 from both sides, ๐‘ฅ+8=5๐‘ฅ+8โˆ’8=5โˆ’88๐‘ฅ=5โˆ’8+8โˆ’8๐‘ฅ=โˆ’35โˆ’8.(subtractfrombothsides)(cancelandontheleft-handside)(calculateontheright-handside)

So, the solution is ๐‘ฅ=โˆ’3.

Further Comments

Notice that, in this example, we got an answer for ๐‘ฅ that was a negative number.

To check that this is correct, we can substitute ๐‘ฅ=โˆ’5 into the expression ๐‘ฅ+8 and check that โˆ’3+8=5. A good way to do this is to use a number line.

Now, let us look at some examples that use multiplication and division.

Example 5: Solving One-Step Linear Equations Involving Division

Solve ๐‘ฅ3=5.

Answer

Remember that ๐‘ฅ3 means ๐‘ฅรท3. So, to get ๐‘ฅ on its own on one side of the equation, we need to use the reciprocal operation of division to cancel out รท3. We can do this by multiplying both sides by 3.

When we do this, the ร—3 cancels with รท3 on the left-hand side, and on the right-hand side we have to calculate 5ร—3.

When we do this, we find that the solution is ๐‘ฅ=15.

Further Comments

Another way to see that multiplying by 3 cancels out the division by 3, think of 3 as being 3รท1 or 31. Then, ๐‘ฅ3ร—3=๐‘ฅ3ร—31.

Now that we have our calculation in fraction form (remember that a fraction is just another way to represent division), we can cancel what is on the top and bottom as long as they are divisible by the same number.

To check the answer, we can verify whether 153=5, which it does.

This shows that ๐‘ฅ3ร—3 is the same as ๐‘ฅ1 which is just equal to ๐‘ฅ.

Example 6: Solving One-Step Linear Equations Involving Multiplication

Solve 4๐‘ฅ=12.

Answer

Remember that 4๐‘ฅ means 4ร—๐‘ฅ. So, to cancel the ร—4 from the left-hand, side we need to use the reciprocal operation of multiplication, which is division. We do this by dividing both sides by 4: 4๐‘ฅ=124๐‘ฅ4=124๐‘ฅ=3.(dividebothsidesby4)

Further Comments

Remember to check the answer. 4ร—3 is 12, so our solution of ๐‘ฅ=3 is indeed correct.

Example 7: Solving One-Step Linear Equations Involving Multiplication

Solve 3๐‘ฅ=โˆ’15.

Answer

Remember that 3๐‘ฅ means 3ร—๐‘ฅ. So, to cancel the ร—3 from the left-hand side, we need to use the reciprocal operation of multiplication, which is division. We do this by dividing both sides by 3: 3๐‘ฅ=โˆ’153๐‘ฅ3=โˆ’153.(dividebothsidesby3)

On the left-hand side, we have 3๐‘ฅ3 so we can cancel the top and bottom because they are both divisible by 3. This gives 1๐‘ฅ1 which simplifies to ๐‘ฅ.

On the right-hand side we, have โˆ’153 which is the same as โˆ’153=โˆ’5.

So, we conclude that ๐‘ฅ=โˆ’5.

Further Comments

Remember to check the answer. 3ร—(โˆ’5) is โˆ’15 since 3ร—5=15 and a positive times a negative is a negative number. So, our solution of ๐‘ฅ=โˆ’5 is indeed correct.

Key Points

  • Below is some of the key vocabulary associated with solving one-step equations.
    • Linear Equations: They are equations that give a straight line when plotted on a graph; for example, ๐‘ฅ+2=5,๐‘ฅโˆ’3=7,2๐‘ฅ=10,๐‘ฅ3=4.and
    • One-Step Equations: They are equations that only require one calculation to solve.
    • Reciprocal Operations/Inverse Operations: These are operations that can undo a calculation. The reciprocal operation of adding is subtracting, the reciprocal operation of multiplying is dividing, and vice versa.
    • Solving an Equation: It means finding the value of ๐‘ฅ that makes the equation true.
  • To solve one-step equations,
    1. identify the reciprocal operation:
      for example, to solve ๐‘ฅ+2=5, we have to subtract 2 to cancel adding 2 to ๐‘ฅ, and to solve 4๐‘ฅ=8, we have to divide by 4 to cancel multiplying ๐‘ฅ by 4 ;
    2. do this operation to both sides:
      remember that the equation must always be balanced;
    3. check your answer by substitution.

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