Step by step
The question
Question: for which k do kx − 4y = 6 and 3x − 2y = 1 have no solution?
We call them ① and ②.
No solution means no point (x, y) works for both.
1 Make the y parts the same
To compare the equations, we make the parts of one letter the same.
k is with x, so we use the other letter, y: it has plain numbers in front.
The y parts: −2y in ② and −4y in ①.
−4 divided by −2 is 2.
So −2y times 2 is −4y.
We multiply every part of ② by 2.
We do it on both sides, so they stay equal.
2 times 3x is 6x.
2 times −2y is −4y.
2 times 1 is 2.
We call it ③.
2 Compare ③ and ①
We write ① again, under ③.
If the left sides are the same, both equations talk about the same amount, 6x − 4y.
One amount cannot be 2 and 6 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 x is left, so we can find x.
That gives one solution.
Careful!
We compare with ③, not ②.
② was times 2.
3x times 2 is 6x.
3 Find k
We want no solution.
So the left sides must be the same: kx = 6x.
k times x is 6 times x when k is 6.
So k = 6.
And the numbers, 2 and 6, are different.
Good.
4 Check
Let's check.
We put k = 6 into ①.
Take ① away from ③.
The left sides are the same, so together they make 0.
The numbers: 2 take away 6.
0 = −4 is never true.
So there is no solution.
It works!
On a grid, the two lines look like this.
Same x and y parts means the same slope.
So the lines are parallel.
Parallel lines never meet: no solution.
Let's look back.
First: make the y parts the same.
Then: compare ③ and ①.
Then: find k.
Last: check.