2x + 3y = 6 and 4x + ky = 5 have no solution. Find k. Easy

The question

Question: for which k do 2x + 3y = 6 and 4x + ky = 5 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: 2x in ① and 4x in ②.

4 divided by 2 is 2.

So 2x times 2 is 4x.

We multiply every part of ① by 2.

We do it on both sides, so they stay equal.

2 times 2x is 4x.

2 times 3y is 6y.

2 times 6 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, 4x + 6y.

One amount cannot be 12 and 5 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.

3y times 2 is 6y.

3 Find k

We want no solution.

So the left sides must be the same: ky = 6y.

k times y is 6 times y when k is 6.

So k = 6.

And the numbers, 12 and 5, are different.

Good.

4 Check

Let's check.

We put k = 6 into ②.

Take ② away from ③.

The left sides are the same, so together they make 0.

The numbers: 12 take away 5.

0 = 7 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.

▶ Watch it solved

2x + 5y = 3 and 6x + ky = 9 have infinitely many solutions. Find k. Medium

The question

Question: for which k do 2x + 5y = 3 and 6x + ky = 9 have infinitely many solutions?

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: 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 3 is 9.

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 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 3.

5y times 3 is 15y.

3 Find k

We want infinitely many solutions.

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, 9 and 9, are the same too.

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: 9 take away 9.

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.

Let's look back.

First: make the x parts the same.

Then: compare ③ and ②.

Then: find k.

Last: check.

▶ Watch it solved

Kx - 4y = 6 and 3x - 2y = 1 have no solution. Find k. Harder

The question

Question: for which k do kx − 4y = 6 and 3x − 2y = 1 have no solution?

We call them ① and ②.

No solution means no point (x, y) works for both.

1 Make the y parts the same

To compare the equations, we make the parts of one letter the same.

k is with x, so we use the other letter, y: it has plain numbers in front.

The y parts: −2y in ② and −4y in ①.

−4 divided by −2 is 2.

So −2y times 2 is −4y.

We multiply every part of ② by 2.

We do it on both sides, so they stay equal.

2 times 3x is 6x.

2 times −2y is −4y.

2 times 1 is 2.

We call it ③.

2 Compare ③ and ①

We write ① again, under ③.

If the left sides are the same, both equations talk about the same amount, 6x − 4y.

One amount cannot be 2 and 6 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 x is left, so we can find x.

That gives one solution.

Careful!

We compare with ③, not ②.

② was times 2.

3x times 2 is 6x.

3 Find k

We want no solution.

So the left sides must be the same: kx = 6x.

k times x is 6 times x when k is 6.

So k = 6.

And the numbers, 2 and 6, are different.

Good.

4 Check

Let's check.

We put k = 6 into ①.

Take ① away from ③.

The left sides are the same, so together they make 0.

The numbers: 2 take away 6.

0 = −4 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 y parts the same.

Then: compare ③ and ①.

Then: find k.

Last: check.

▶ Watch it solved