Variable Equations

An equation is a math sentence, says that two things are equal to each other.

One of the most important math concepts is solving equations such as x + 3 = 5.

This is called a variable equation.

In this equation, x is a variable.

The goal when solving equations is to figure out what the variable equals.

In other words, what number can you put in place of the variable that will make the statement true?

In order to do that, do the opposite operation to each side.

That means that if a number is added, you must subtract (and vice versa). If a number is multiplied, you must divide (and vice versa).

Let's look at a few examples to see how to do that.

Examples 1

1.  9 + n = 10

This equation says that 9 plus something (n) equals 10.

To find out, we must get n by itself.

What is on the same side as the n?

Since it's being added, we must do the opposite - subtraction.

We must subtract 9 from both sides of the equation.

9 + n = 10

9 + n - 9 = 10 - 9

n = 1            [On the left, we get just n since 9 - 9 = 0. On the right, 10 - 9 = 1]

n = 1

We know this is right because it makes the equation true. 9 + 1 = 10.

2.  4 x n = 32

This equation says that 4 times something (n) equals 32.

Since, 4 is being multiplied, we must divide both sides by 4.

4 x n = 32

1 x n = 8            [On the left, 4 ÷ 4 = 1 and 1 x n = n.On the right, 32 ÷ 4 = 8.]

n = 8

We know this is right because 4 x 8 = 32.

3.  2x + 4 = 10

This equation has two things that need to be moved - the 2 and the 4.

You must always move the number being added or subtracted first.

2x + 4 = 10

2x + 4 - 4 = 10 - 4            [On the left, 4 - 4 = 0 and 2x is still there. On the right, 10 - 4 = 6]

2x = 6

Divide both sides by 2.

2x = 6

2x ÷ 2 = 6 ÷ 2

x = 3

We can check our answer by putting 3 back into the original equation in place of x.

2x + 4 = 10

2(3) + 4 = 10.

This is true, so we know we are correct.

4.  2(x + 6) = 20

In the last problem, we learned that you have to move the number that's being added or subtracted before the number that's being multiplied or divided.

But this problem is an exception to that rule because something is different.

Do you see what it is?

It's because of the parenthesis.

Parenthesis are grouping symbols that alert us that what's inside is being grouped together.

Normally this would mean we should add x + 6 first, but since we don't know what x is we can't do that. Instead, we're trying to get x by itself.

And since x + 6 is a group, we have to move the 2 first.

Since the 2 is being multiplied

Remember : A number right next to parenthesis with nothing in between means multiplication.

We must divide both sides by 2.

2(x + 6) = 20            [ On the left, 2 ÷ 2 = 1 and just x + 6 is left ]

(x + 6) = 10

The parenthesis are no longer needed since x + 6 is the only thing on that side of the equation.

Now we can move 6 by subtracting 6 from both sides.

x + 6 = 10

x + 6 - 6 = 10 - 6

x = 4

We can check if this is true by putting 4 back in for x.

2(4 + 6) = 20

Remember to do what's in parenthesis first.

2(10) = 20

Word problems -

Examples 1

Joe practiced the piano for the same amount of time every day last week. He practiced for 70 minutes total. How much did he practice each day?

To solve this, we can first write an equation. We don't know how long he practiced each day, so that will have to be a variable.

Let's use 't' for time.

We know he practiced for 7 days (because there are 7 days in a week.)

So, 7 times t (the amount he practiced each day) must equal 70 minutes.

7 x t = 70

We can now solve this equation.

7 times what number equals 70?

10 because 7 x 10 = 70

t = 10

So, Joe practiced 10 minutes each day.

Examples 2

Alexandra bought 3 more pieces of candy this week than she did last week. This week she bought 8 pieces. How many did she buy last week?

To write an equation, we need to figure out what we are looking for. We want to find out how

many pieces she bought last week. We don't know this, so it will be our variable.

Let's use 'c' for candy.

Alexandra bought 3 more candies this week than last week. So, the sum of 'c' and 3 must be 8.

3 + c = 8

Now we can solve it by thinking 3 plus what number equals 8?

3 + 5 = 8

Here, c = 5

So, Alexandra bought 5 pieces of candy last week.