Step by step
The question
Question: for which k do x + y = 7 and x + ky = 2 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: x in ① and x in ②.
They are already the same.
So we can compare ① and ② straight away.
2 Compare ① and ②
If the left sides are the same, both equations talk about the same amount, x + y.
One amount cannot be 7 and 2 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.
3 Find k
We want no solution.
So the left sides must be the same: ky = y.
k times y is 1 times y when k is 1.
So k = 1.
And the numbers, 7 and 2, are different.
Good.
4 Check
Let's check.
We put k = 1 into ②.
Take ② away from ①.
The left sides are the same, so together they make 0.
The numbers: 7 take away 2.
0 = 5 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 ① and ②.
Then: find k.
Last: check.