Post 12: Substitution as a Solving Strategy

(by Mark O. with help from Sage the intrepid student dog)

A shapeshifting creature blocks the doorway, half-melted and lopsided, above which two equations float. The shadowy figure had been here first, trying to use one equation to solve the other, but didn't quite do it right. This left the creature mismatched and strange. A small cluster of kids drifts closer to watch as Sage tries to help the creature.

As a reminder, substitution is a method for solving a system of equations with two variables. First, you isolate one variable, getting it alone on one side of one equation. Then you take what that variable equals and substitute it into the other equation, replacing that same variable there.

Let's look at a way that people might make mistakes when substituting equations:

Problem: Forgetting to substitute everywhere

When a variable shows up more than once in an expression, it's easy to swap in a value for the first one you see and stop there. But every x in the expression has to become the same number, or the pieces stop matching each other.

Which of these correctly substitutes x = 4 into 2x + 3x − 1?

A) 2(4) + 3x − 1 B) 2(4) + 3(4) − 1

(It's B.)

Sage sees this set of equations above the creature:

He tries to solve the system like this, but there's a mistake. Can you help fix it? (Just click on the math and start typing to edit. Hold control and press enter to add new lines).

It can be really easy to accidentally substitute the wrong equation. Always make sure that you are substituting the right variable!

Let's finish this system to help the creature!

Practice this one on your own! (Hold control and press enter to add more math lines):

As Sage finishes the last substitution, the creature settles all at once into a single coherent shape, calm and clearly friendly. The creature gently opens the door, and Sage leads the group of kids through the door.

Catch your algebra mistakes before they cost you

The scratchpad in this guide runs on moment.of.math — a free Chrome extension that checks your algebra line by line, on any problem. It can generate new variations problems, so you can get as much practice as you need. On many homework sites, it automatically detects the math on the page, so you don't have to copy anything over. Try it on tonight's homework.

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