Step by step
The question
Question: for which k do 2x + 5y = 4 and 6x + ky = 7 have no solution?
Two equations, two letters, and one number k to find.
There are four choices.
First we solve it, the normal way.
Then we pick.
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 6x in ②.
6 divided by 2 is 3.
So 2x times 3 is 6x.
We multiply every part of ① by 3.
We do it on both sides, so they stay equal.
3 times 2x is 6x.
3 times 5y is 15y.
3 times 4 is 12.
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, 6x + 15y.
One amount cannot be 12 and 7 at once.
So no point works: no solution.
Careful!
We compare with ③, not ①.
① was times 3.
5y times 3 is 15y.
3 Find k
We want no solution.
So the left sides must be the same: ky = 15y.
k times y is 15 times y when k is 15.
So k = 15.
And the numbers, 12 and 7, are different.
Good.
4 Check
Let's check.
We put k = 15 into ②.
Take ② away from ③.
The left sides are the same, so together they make 0.
The numbers: 12 take away 7.
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.
5 Pick the choice
Our answer is k = 15.
That is choice C.
6 Why not the others
A is 5.
Wrong: kept the y number: but the x number tripled, 2 → 6.
That is the Careful mistake from before.
B is 10.
Wrong: doubled: 2 → 6 is times 3, not times 2.
D is 35.
Wrong: 5 × 7: used the 7, not the 3 from 2 → 6.
Let's look back.
First: make the x parts the same.
Then: compare ③ and ②.
Then: find k.
Then: check.
Then: pick the choice.
Last: why not the others.