Step by step
The question
Question: for which k do 2x − 3y = 1 and −4x + ky = 7 have no solution?
We call them ① and ②.
No solution means no point (x, y) works for both.
1 Make the x parts the same
To compare the equations, we make the parts of one letter the same.
k is with y, so we use the other letter, x: it has plain numbers in front.
The x parts: 2x in ① and −4x in ②.
−4 divided by 2 is −2.
So 2x times −2 is −4x.
We multiply every part of ① by −2.
We do it on both sides, so they stay equal.
−2 times 2x is −4x.
−2 times −3y is 6y.
−2 times 1 is −2.
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, −4x + 6y.
One amount cannot be −2 and 7 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.
−3y times −2 is 6y.
3 Find k
We want no solution.
So the left sides must be the same: ky = 6y.
k times y is 6 times y when k is 6.
So k = 6.
And the numbers, −2 and 7, are different.
Good.
4 Check
Let's check.
We put k = 6 into ②.
Take ② away from ③.
The left sides are the same, so together they make 0.
The numbers: −2 take away 7.
0 = −9 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 x parts the same.
Then: compare ③ and ②.
Then: find k.
Last: check.