Question

Which of these statements is correct?

The system of linear equations 6x - 5y = 8 and 12x - 10y = 16 has no solution.

The system of linear equations 7x + 2y = 6 and 14x + 4y = 16 has an infinite number of solutions.

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

The system of linear equations 9x + 6y = 14 and 18x + 12y = 26 has an infinite number of solutions.

Answer 1

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

Explanation:

The true statement is the third one, because that system of equations has no solutions. This is because those lines are parallel, see image attached.

We can demonstrate this by solving the system:

`\left \{ {{8x-3y=10} \atop {16x-6y=22}} \right.`

If we multiply the first equation by -2, we would have

`\left \{ {{-16x+6y=-20} \atop {16x-6y=22}} \right\\0x+0y=2\\0=2`

When this happens, means that the system has no solution, that is, the lines that represents those linear equations, are parallel.

Therefore, the right answer is the third option.

Answer 2

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

Explanation:

1) The system of linear equations 6x - 5y = 8 and 12x - 10y = 16 has no solution.

Solve these linear equations simultaneously

Step 1 : Find y in terms of x from any one equation

6x - 5y = 8

y = 8 - 6x

-5

Step 2 : Substitute y in terms of x from step 1 in the second equation.

16x - 6y = 22

16x - 6 (8 - 6x) = 22

-5

80x - 48 + 36x = 22 x -5

94x = 43

x = 0.457

This statement is incorrect as it does have a solution.

2) The system of linear equations 7x + 2y = 6 and 14x + 4y = 16 has an infinite number of solutions.

Solve these linear equations simultaneously

Step 1 : Find y in terms of x from any one equation

7x + 2y = 6

y = 6 - 7x

2

Step 2 : Substitute y in terms of x from step 1 in the second equation.

14x + 4y = 16

14x + 4(6 - 7x) = 16

2

14x + 12 - 14x = 16

0 ≠ 4

This statement is not true as there are no solutions.

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

Solve these linear equations simultaneously

Step 1 : Find x in terms of y from any one equation

8x - 3y = 10

x = 10 + 3y

8

Step 2 : Substitute x in terms of y from step 1 in the second equation.

16x - 6y = 22

16(10 + 3y) - 6y = 22

8

20 + 6y - 6y = 2

0 ≠ -18

This statement is true because there are no solutions

4) The system of linear equations 9x + 6y = 14 and 18x + 12y = 26 has an infinite number of solutions.

Solve these linear equations simultaneously

Step 1 : Find x in terms of y from any one equation

9x + 6y = 14

x = 14 - 6y

9

Step 2 : Substitute x in terms of y from step 1 in the second equation.

18x + 12y = 26

18 (14 - 6y) + 12y = 26

9

8 - 12y + 12y = 26

0 ≠ 18

This statement is incorrect because there are no solutions. It does not have infinite number of solutions.

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