1

Problem

Show that the following system of equations has an infinite solution:

-14x + 21y = 35 and 2x - 3y = - 5

2

Step

Given system of the equations are

-14x + 21y = 35 Equation(1)

2x - 3y = - 5 Equation(2)

3

Hint

The linear system is
\$ a_1x + b_1y = c_1 \$

\$ a_2x + b_2y = c_2 \$

4

Step

By comparing with the linear system, then simplify:

\$ a_1 = - 14, b_1 = 21 ,c_1 = 35 , a_2 = 2 b_2 = - 3,c_2 = - 5 \$

5

Step

Now, the ratios are:

\$ (a_1/a_2) = (\cancel(-14)^7 )/(\cancel(2)^1 ) = - 7 \$

\$ (b_1/b_2) = (\cancel(21)^7 )/(\cancel(- 3)^1 ) = - 7 \$

\$ (c_1/c_2) = (\cancel(35)^7 )/(\cancel(- 5)^1 ) = - 7 \$

\$ (a_1/a_2) = (b_1/b_2) = (c_1/c_2) \$

6

Solution

Therefore, the given system of equations has infinitely many solutions.

7

Sumup

Please summarize steps

Choices

8

Choice-A

This option indicates that the ratios of coefficients are not equal, which is not true in this case

Wrong \$- 7 ne 7 ne - 7 \$

9

Choice-B

This option indicates that the ratios of coefficients are not equal, which is not true in this case

Wrong \$ - 1/7 ne - 7 = - 7\$

10

Choice-C

This option indicates that the ratios of coefficients are equal, but this is not true

Wrong 5 = 5 = 5

11

Choice-D

This option indicates that all the ratios of coefficients are equal, which is true in this case. So, this option is correct

Correct - 7 = - 7 = - 7

12

Answer

Option

D

13

Sumup

Please summarize choices