Solving Equations

In this example, we are asked to solve for x in the equation five x plus twelve is equal to seven x plus negative eight.

This will be a crowded setup when I represent the equation on the mat using cups and cubes. Starting with the left side of the equation, five x plus twelve, I will need to place five cups and twelve green cubes. And for the right side of the equation, seven x plus negative eight, I will place seven cups and eight black cubes.

And now the trial and error strategy. Time out. There's gotta be a more efficient way.

And in fact, there is, and it's called balanced moves. And just like the name suggests, we will make moves to make the problem simpler, all while keeping the scale perfectly balanced.

Our objective here is to isolate the variable, x in this case, which simply means we want to have the cups on only one side. But we can't just simply move the cups or the scale will become unbalanced.

So what I will do is remove a cup from each side of the equation at the same time. All of the cups have the same value, so the equation will stay completely balanced. When you do this, use two hands removing one cup from each side.

Continue the process of removing a cup from each side until you have successfully isolated the variable. Meaning, only one side of the equation has any cups.

Now this equation looks a lot more manageable to solve. Rather than looking at that complicated equation at the start, after using balanced moves, the equation has been simplified. Under the left side, there are just twelve cubes. And on the right side, there are two cups and negative eight cubes. Twelve equals two x plus negative eight.

And rather than jumping straight into trial and error to solve for x, I will first use estimation. I know that the value of the two cups and negative eight cubes must be equal to twelve. So I can think, what must the value of the cups be so that when I take away eight, I'm left with twelve? I know that twenty minus eight equals twelve.

So let me test my thinking by putting ten cubes into each cup. If each cup is equal to ten, the left side of the equation still remains twelve, and the right side of the equation is ten twenty take away eight. That equals twelve.

The scale is balanced when the value of x is ten.

We can check this mathematically by substituting ten in for x back into the original equation. Five times ten plus twelve equals seven times ten plus negative eight. I'll do the multiplication first. Sixty two equals sixty two. It checks out.

If you find yourself in a situation where you are asked to solve this type of problem using pencil and paper, no worries, you can still do it. Draw large circles for each x, small squares to represent the positive numbers, and shaded in squares for negative numbers. Then use balanced moves crossing out one cup at a time on each side. Then test different values for the cups until both sides are balanced.

And now you're ready to solve equations by using balanced moves to isolate the variable.