Step by step
The question
Question: for which k do 3x + 2y = 7 and 6x + ky = 5 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: 3x in ① and 6x in ②.
6 divided by 3 is 2.
So 3x times 2 is 6x.
We multiply every part of ① by 2.
We do it on both sides, so they stay equal.
2 times 3x is 6x.
2 times 2y is 4y.
2 times 7 is 14.
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 + 4y.
One amount cannot be 14 and 5 at once.
So no point works: no solution.
Careful!
We compare with ③, not ①.
① was times 2.
2y times 2 is 4y.
3 Find k
We want no solution.
So the left sides must be the same: ky = 4y.
k times y is 4 times y when k is 4.
So k = 4.
And the numbers, 14 and 5, are different.
Good.
4 Check
Let's check.
We put k = 4 into ②.
Take ② away from ③.
The left sides are the same, so together they make 0.
The numbers: 14 take away 5.
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.
5 Pick the choice
Our answer is k = 4.
That is choice C.
6 Why not the others
A is −4.
Wrong: the sign: 2 × 2 is +4.
B is 2.
Wrong: kept the y number: but the x number doubled, 3 → 6.
That is the Careful mistake from before.
D is 6.
Wrong: copied the x number 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.