# MPM2D Substitution method – special cases – 2016-09-16

We’ve practised finding intersections of lines by substitution when we had a variable isolated already in at least one equation, like $y=3x-9$.

Today we learned how to deal with cases in which neither equation has a variable isolated, such as $3x-2y+6=0$ and $x+4y+1=0$.

The two strategies we have so far are:

1. Convert one equation into Slope-y-intercept form, then substitute (this is essentially isolating for $y$
2. Isolate one equation for $x$ and then substitute (same thing, except it’s for $x$)

We’ll learn another technique next week.

## The Three Possible Outcomes

When you have two linear equations, three things are possible.

• The two equations represent two lines that are not parallel. When substituting, you’ll be able to solve for $x_1$ or $y_1$ and get an actual number, like $x_1=\frac{4}{7}$.
• The two equations represent parallel lines that are distinct (they’re not the same line). When substituting you’ll eventually get an equation that cannot be true (such as $-6=0$.
• The two equations represent parallel lines that are identical (they’re the same line). When substituting you’ll eventually get an equation that is always true (such as $5=5$.

## The Homework

Complete textbook questions 1 and 2 on Page 26. For both questions please also solve for the intersection points.