Step by step
The question
Question: for which k do 3x + y = 4 and 12x + ky = 16 have infinitely many solutions?
Two equations, two letters, and one number k to find.
We call them ① and ②.
Infinitely many solutions means every point on a line 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 12x in ②.
12 divided by 3 is 4.
So 3x times 4 is 12x.
We multiply every part of ① by 4.
We do it on both sides, so they stay equal.
4 times 3x is 12x.
4 times y is 4y.
4 times 4 is 16.
We call it ③.
2 Compare ③ and ②
We write ② again, under ③.
If the left sides are the same and the numbers are the same, it is one equation written twice.
Every point on that line works: infinitely many solutions.
Careful!
We compare with ③, not ①.
① was times 4.
y times 4 is 4y.
3 Find k
We want infinitely many solutions.
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, 16 and 16, are the same too.
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: 16 take away 16.
0 = 0 is always true.
They are the same equation.
It works!
On a grid, both equations give the same line.
② lies right on top of ①.
Every point on the line works for both: infinitely many solutions.
5 Type it in
The answer box takes 5 characters.
4 has 1.
It fits.
We type 4.
Let's look back.
First: make the x parts the same.
Then: compare ③ and ②.
Then: find k.
Then: check.
Last: type it in.