Step by step
The question
Question: for which k do 4x − y = 2 and 8x + ky = 3 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: 4x in ① and 8x in ②.
8 divided by 4 is 2.
So 4x times 2 is 8x.
We multiply every part of ① by 2.
We do it on both sides, so they stay equal.
2 times 4x is 8x.
2 times −y is −2y.
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, 8x − 2y.
One amount cannot be 4 and 3 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.
−y times 2 is −2y.
3 Find k
We want no solution.
So the left sides must be the same: ky = −2y.
k times y is −2 times y when k is −2.
So k = −2.
And the numbers, 4 and 3, are different.
Good.
4 Check
Let's check.
We put k = −2 into ②.
Take ② away from ③.
The left sides are the same, so together they make 0.
The numbers: 4 take away 3.
0 = 1 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.