Examples of Solutions

Today, we're going to explore the solutions of linear equations in two variables. Can anyone tell me what a solution actually means?

I think it's the values that make the equation true?

Exactly! When we have an equation like $2x + 3y = 12$, a solution is any pair of values for $x$ and $y$ that satisfies this equation.

So, can we find multiple pairs that work?

Yes! In fact, for linear equations in two variables, there are infinitely many solutions.

Let's take the equation $2x + 3y = 12$. Can anyone suggest a value for $x$ that we might use?

What if we try $x = 0$?

Great choice! Plugging $x = 0$ in, we get $3y = 12$, which gives us $y = 4$. So one solution is $(0, 4)$.

Can we find another pair using a different value?

Absolutely! If we choose $x = 3$, then $2(3) + 3y = 12$ leads us to $y = 2$. Another solution is $(3, 2)$.

Now, seeing how we can substitute different values generates new solutions. Who would like to suggest another value?

How about we set $x = 6$?

Excellent! Plugging $x = 6$ into the equation gives us $2(6) + 3y = 12$, leading to $y = 0$. Thus, we have another solution: $(6, 0)$.

So we can get all sorts of points just by changing $x$!

Exactly! And this is just one way to find solutions. You can start with either variable!

Now, let’s verify if certain pairs are indeed solutions for the equation $x - 2y = 4$. Who can help me check $(2, 0)$?

I think we need to plug in $x = 2$ and $y = 0$. So, $2 - 0 = 4$ is false.

Correct! That pair does not work. What about $(4, 0)$?

For $4 - 2(0) = 4$, that works!

Great job! Validating solutions helps us identify which pairs are feasible.

Let's summarize! What have we learned about solutions for linear equations in two variables?

They can have many solutions, often represented as pairs!

You can find solutions by choosing values for either $x$ or $y$!

Exactly, and validating each pair ensures accuracy. Well done everyone!