Step by step
The question
Question: for which k do 5x + 2y = 10 and 10x + 4y = k 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.
Both letters have plain numbers in front.
We start with x.
The x parts: 5x in ① and 10x in ②.
10 divided by 5 is 2.
So 5x times 2 is 10x.
We multiply every part of ① by 2.
We do it on both sides, so they stay equal.
2 times 5x is 10x.
2 times 2y is 4y.
2 times 10 is 20.
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, 10x + 4y.
One amount cannot be two different numbers 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.
10 times 2 is 20.
3 Find k
The left sides, 10x + 4y, are already the same.
So the numbers must be different.
k can be any number except 20.
4 Check
Let's check with one k that is not 20.
We pick k = 21.
Take ② away from ③.
The left sides are the same, so together they make 0.
The numbers: 20 take away 21.
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.