Question

Which of these statements is correct? The system of linear equations 6 x minus 5 y = 8 and 12 x minus 10 y = 16 has no solution. The system of linear equations 7 x + 2 y = 6 and 14 x + 4 y = 16 has an infinite number of solutions. The system of linear equations 8 x minus 3 y = 10 and 16 x minus 6 y = 22 has no solution. The system of linear equations 9 x + 6 y = 14 and 18 x + 12 y = 26 has an infinite number of solutions

Answer 1

The answer to your question is the third choice.

Explanation:

a) 6x - 5y = 8

12x - 10y = 16

We observe that these lines are the same so they have infinite solutions.

b)

7x + 2y = 6

14x + 4y = 16

These lines are parallel because they have the same slope, so they do not cross, there is no solution.

c)

8x - 3y = 10

16x - 6y = 22

These lines are parallel because they have the same slope, so they do not cross, there is no solution.

d)

9x + 6y = 14

18x + 12y = 26

These lines are parallel because they have the same slope, so they do not cross, they do not have an infinite number of solutions.

Answer 2

The only true statement is:

"The system of linear equations 8x - 3y = 10 and 16x - 6y = 22 has no solution."

Explanation:

First, some definitions.

A system of linear equations has infinite solutions if both equations define the same line, has no solutions if we have two parallel lines, has one solution in all the other cases.

Where two lines are parallel if we can write them as:

a*x + b*y = c

a*x + b*y = d

where c and d are different numbers.

Now we can analyze the given statements:

a)

6x - 5y = 8

12x - 10y = 16

has no solution?

If we divide both sides of the second equation by 2, we get:

(12x - 10y)/2 = 16/2

6x - 5y = 8

We get the first equation, then both equations define the same line, thus the system has infinite solutions, then the statement is false.

b)

7x + 2y = 6

14x + 4y = 16

has infinite solutions?

Let's divide the second equation by 2, then we get:

(14x + 4y)/2 = 16/2

7x + 2y = 8

If we rewrite our system of equations, we get:

7x + 2y = 6

7x + 2y = 8

These are parallel lines, thus, this system has no solutions.

So the statement is false.

c)

8x - 3y = 10

16x - 6y = 22

has no solution?

Again, let's divide the second equation by 2 to get:

(16x - 6y)/2 = 22/2

8x - 3y = 11

If we rewrite our system:

8x - 3y = 10

8x - 3y = 11

These are parallel lines, thus the system has no solutions, so this statement is correct.

d)

9x + 6y = 14

18x + 12y = 26

Has infinite solutions?

Dividing the second equation by 2 we get:

(18x + 12y)/2 = 26/2

9x + 6y = 13

So the equations are different (are parallel lines again) so this system has not infinite solutions.

Then the statement is false.