Step by step
The question
Question: for which k do kx + 6y = 4 and 2x + 3y = 5 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: 3y in ② and 6y in ①.
6 divided by 3 is 2.
So 3y times 2 is 6y.
We multiply every part of ② by 2.
We do it on both sides, so they stay equal.
2 times 2x is 4x.
2 times 3y is 6y.
2 times 5 is 10.
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, 4x + 6y.
One amount cannot be 10 and 4 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.
2x times 2 is 4x.
3 Find k
We want no solution.
So the left sides must be the same: kx = 4x.
k times x is 4 times x when k is 4.
So k = 4.
And the numbers, 10 and 4, are different.
Good.
4 Check
Let's check.
We put k = 4 into ①.
Take ① away from ③.
The left sides are the same, so together they make 0.
The numbers: 10 take away 4.
0 = 6 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.