Substitution

In this lesson you will: Isolate one variable in one equation, then put that expression into the other equation; The result is a single equation in one variable — Chapter 3 work; Back-substitute to get the second coordinate, then check the pair in both originals.

Trade one equation for an expression

If y=2x+1, then anywhere the letter y appears you are entitled to write 2x+1 instead. They name the same number. That is the whole engine of substitution.

Take the system y=2x+1 and 3x+y=11. Replace y in the second equation: 3x+(2x+1)=11. Now there is one variable, and you already know how to finish. 5x+1=11, so x=2, and then y=2(2)+1=5.

Choose the variable that is cheapest to isolate

Not every system hands you a y=.... When none is given, look for a variable whose coefficient is 1 or -1 and solve for that one. Isolating it costs no fractions.

In 2x+y=7 and 3x-2y=7, the y in the first equation has coefficient 1. Solve that one: y=7-2x. Isolating x in the second instead would give x=(7+2y)/(3), and you would carry a fraction through every line that follows.

Finish the job

Solving for x is not the answer. A system asks for a pair, so take the value you found and put it back into the simplest equation you have — usually the one you rearranged — to get the other coordinate.

Then check the pair in both original equations. That last step catches a sign error in about ten seconds.

Common mistake — Substituting back into the equation you came from

Given y=4-x and 2x+y=9, a student solves the first for y, then substitutes 4-x into the first equation and writes 4-x=4-x.

Nothing looks wrong while it is happening. The algebra is valid, the line is true, and the student is doing exactly what the word \"substitute\" says.

But an equation cannot tell you anything new about itself. Putting y=4-x back into y=4-x produces a statement that is true for every x, which is why the variable vanishes and no progress is made.

The expression has to travel to the other equation, the one that has not been used yet: 2x+(4-x)=9, so x+4=9 and x=5, giving y=-1.

How to catch it: Put a small mark beside the equation you took the expression from, and never substitute into a marked equation. If your substitution produces something like 4-x=4-x or 9=9 on the first move, you have used the same equation twice.