Equations with Negative Numbers

In this example, we are given an equation that uses a negative number. X plus eleven is equal to 3x plus negative five.

Let's take a look at how to set up and solve this equation on our mat using cups and cubes.

Starting with the left side of the equation, x plus eleven, I will use one cup to represent the variable x, and then eleven cubes.

Now for the right side of the equation, 3x plus negative five. There's nothing new about the three x. That just means three cups. But how do I represent negative five? There's no way to show a negative cube. Or is there?

To represent negative values, use black cubes. Green cubes for positive numbers and black cubes for negative numbers. To show negative five, I will place five black cubes on the equations mat.

And now I'm ready to solve this equation using the trial and error strategy. I will start by placing one green cube in each cup.

If each cup has a value of one, then the left side of the equation is equal to twelve. And the right side of the equation? Well, that's a little trickier.

The cups total three, and then I need to add negative five. Let's explore how to do that. Adding a negative number is just like subtracting a positive number. Three plus negative five is the same as three minus five.

However we think about it, we need to start at three and count back five. And that gets us to a number less than zero. Into the world of negative numbers, it's negative two.

When each cup has a value of one, the equation is not even close to balanced. I'm going to try adding a bunch more green cubes to each cup.

Now each cup has a value of five, and the left side is equal to sixteen. The right side is five, ten, fifteen, plus negative five. Whether we think about it using subtraction or counting back five, the total is ten. Sides are getting closer, but they still aren't balanced.

I'm going to add three more cubes to each cup. When each cup has a value of eight, the left side is equal to nineteen, and the right side is eight, sixteen, twenty four plus negative five. That's also nineteen!

We have an answer. The value of each cup is eight.

We can check this mathematically by substituting eight into the equation for x. Eight plus eleven equals eight plus eight plus eight plus negative five. Nineteen equals nineteen. It checks out.

And if you find yourself in a situation where you're asked to solve this type of problem using pencil and paper, no worries, you can still do it. Draw large circles for x, small squares to represent the positive numbers, and shaded in squares for negative numbers. Then use the trial and error strategy, testing different values until both sides are balanced.

And now you're ready to begin solving equations that use negative numbers.