Introduction

Welcome class! Today, we will explore linear equations in two variables. Can anyone recall what we learned about linear equations in one variable?

Yes! A linear equation in one variable can have one unique solution.

Correct! Now, if we have two variables, like $x$ and $y$, what might we expect in terms of solutions?

Maybe it has more than one solution since there are two unknowns!

Exactly! In fact, a linear equation in two variables can have infinitely many solutions, represented as points in a 2D space.

Can we use the Cartesian plane for these solutions?

Yes! Every solution is a point on the Cartesian plane, and we can visualize this easily.

What does the general form of these equations look like?

Great question! The general form is $ax + by + c = 0$, where $a, b, c$ are constants. Let’s break that down.

Let’s look at some examples. How can we convert $2x + 3y = 4$ into the standard form?

We can rearrange it to $2x + 3y - 4 = 0$.

Exactly! Here, $a = 2$, $b = 3$, and $c = -4$. Who can do another one?

How about $x - 5 = 3y$? It becomes $x - 3y - 5 = 0$!

Well done! Each of these forms helps us identify the coefficients. Can someone explain why this matters?

Because we need to understand the relationship between the variables!

Yes! This relationship is crucial for solving equations.

Next, let’s discuss solutions. If we take the equation $2x + 3y = 12$, how can we find a solution?

We can substitute values for $x$ and solve for $y$!

Exactly! For instance, if $x = 0$, what do we get for $y$?

That would result in $y = 4$, giving us the solution $(0, 4)$.

Great job! And what happens if we try $x = 3$?

Then $3y = 6$ and $y = 2$, leading to another solution $(3, 2)$.

Exactly! This process can be repeated to find infinitely many solutions.