Question

Which of these systems of linear equations has no solution? 2 x + 8 y = 15. 4 x + 16 y = 30. 2 x minus y = 18. 4 x + 2 y = 38. 4 x + 7 y = 17. 8 x minus 14 y = 36. 4 x minus 3 y = 16. 8 x minus 6 y = 34.

Answer 1

Answer:

The correct answer is:

$4 x -3 y = 16$ and

$8 x -6 y = 34$ are the equations that have no solution.

Step-by-step explanation:

First of all, let us have a look at the rules for a pair of lines, that have solution or no solution.

Let the equations be:

$a_1x+b_1y+c_1=0$ and

$a_2x+b_2y+c_2=0$

1. There exists exactly one solution:

$\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}$

2. There exists infinitely many solutions i.e. lines are identical.

$\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}$

3. There exists No solutions i.e. lines are parallel and will never intersect.

$\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}$

Now, let us have a look at the given pair of lines:

Option a)

$2 x + 8 y = 15\\4 x + 16 y = 30.$

The ratio:

$\dfrac{2}{4}=\dfrac{8}{16}=\dfrac{15}{30} = \dfrac{1}{2}$

Hence, the lines are identical, infinite solutions.

Option b)

$2 x - y = 18\\4 x + 2 y = 38$

The ratio:

$\dfrac{2}{4}\neq \dfrac{-1}{2}$

Hence, exactly one solution.

Option c)

$4 x + 7 y = 17\\8 x -14 y = 36$

The ratio:

$\dfrac{4}{8}\neq \dfrac{-7}{14}\\\dfrac{1}{2}\neq \dfrac{-1}{2}$

Hence, exactly one solution.

Option d)

$4 x - 3 y = 16\\8 x - 6 y = 34.$

The ratio:

$\dfrac{4}{8}= \dfrac{-3}{-6}\neq\dfrac{16}{34}\\\Rightarrow \dfrac{1}{2}= \dfrac{1}{2}\neq\dfrac{8}{17}$

Hence, There is no solution.

The correct answer is:

$4 x -3 y = 16$ and

$8 x -6 y = 34$ are the equations that have no solution.

Answer 2

Answer:

d

Step-by-step explanation: