1

Problem

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

3x - y = 2 and 9x - 3y = 6

2

Step

Given system of the equations are

3x - y = 2 Equation (1)

9x - 3y = 6 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 = 3, b_1 = -1 ,c_1 = 2 , a_2 = 9 b_2 = - 3 ,c_2 = 6 \$

5

Step

Now, the ratios are:

\$ (a_1/a_2) = \cancel (3)^1 / (\ cancel (9)^2 = 1/3 \$

\$ (b_1/b_2) = -1 /-3 = 1/3 \$

\$ (c_1/c_2) = \cancel2^1/(\cancel6^3) = 1/3 \$

\$ (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 is correct because Equation 2 is a multiple of Equation 1, making the ratios of corresponding coefficients equal

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

9

Choice-B

This is incorrect because the coefficients' ratios in Equation 2 are proportional to Equation 1

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

10

Choice-C

This is incorrect because the coefficients ratio in Equation 2 is a multiple of Equation 1

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

11

Choice-D

This is incorrect because Equation 2 is a multiple of Equation 1, indicating unequal coefficients

Wrong \$ (a_1)/(a_2) ne (b_1)/(b_2) ne (c_1)/(c_2) \$

12

Answer

Option

A

13

Sumup

Please summarize choices