Step by step
The question
Question: for which k do x + 3y = 2 and 2x + 6y = k have infinitely many solutions?
We call them ① and ②.
Infinitely many solutions means every point on a line works for both.
1 Make the x parts the same
To compare the equations, we make the parts of one letter the same.
Both letters have plain numbers in front.
We start with x.
The x parts: x in ① and 2x in ②.
2 divided by 1 is 2.
So x times 2 is 2x.
We multiply every part of ① by 2.
We do it on both sides, so they stay equal.
2 times x is 2x.
2 times 3y is 6y.
2 times 2 is 4.
We call it ③.
2 Compare ③ and ②
We write ② again, under ③.
If the left sides are the same, both equations talk about the same amount, 2x + 6y.
One amount cannot be two different numbers at once.
So no point works: no solution.
If the numbers are the same too, it is one equation written twice.
Every point on that line works: infinitely many solutions.
If the left sides are different, we can take one equation away from the other.
Some y is left, so we can find y.
That gives one solution.
Careful!
We compare with ③, not ①.
① was times 2.
2 times 2 is 4.
3 Find k
The left sides, 2x + 6y, are already the same.
So the numbers must be the same too: k = 4.
4 Check
Let's check.
We put k = 4 into ②.
Take ② away from ③.
The left sides are the same, so together they make 0.
The numbers: 4 take away 4.
0 = 0 is always true.
They are the same equation.
It works!
On a grid, both equations give the same line.
② lies right on top of ①.
Every point on the line works for both: infinitely many solutions.
Let's look back.
First: make the x parts the same.
Then: compare ③ and ②.
Then: find k.
Last: check.