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Lesson Example Discussion Quiz: Class Homework

Step-3

Title: System of Linear Equations with Infinite Solutions

Grade: 1400-a Lesson: S1-L7

Explanation: Hello Students, time to practice and review the steps for the problem.

Lesson Steps

Step	Type	Explanation	Answer
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	Formula:	The linear system is	\$ a_1x + b_1y = c_1 \$ \$ a_2x + b_2y = c_2 \$
4	Hint	By comparing with the linear system, we get	\$ 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	Step	Therefore, the given system of equations has infinitely many solutions.
7	Choice.A	This is correct because Equation 2 is a multiple of Equation 1, making the ratios of corresponding coefficients equal	\$ (a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2) \$
8	Choice.B	This is incorrect because the coefficients ratios in Equation 2 are proportional to Equation 1	\$ (a_1)/(a_2) ne (b_1)/(b_2) = (c_1)/(c_2) \$
9	Choice.C	This is incorrect because the coefficients ratio in Equation 2 is a multiple of Equation 1	\$ (a_1)/(a_2) = (b_1)/(b_2) ne (c_1)/(c_2) \$
10	Choice.D	This is incorrect because Equation 2 is a multiple of Equation 1, indicating unequal coefficients	\$ (a_1)/(a_2) ne (b_1)/(b_2) ne (c_1)/(c_2) \$
11	Answer	Option	A
12	Sumup	Can you summarize what you’ve understood in the above steps?

Step

Type

Explanation

Answer

Problem

Show that the following system of equations has an infinite solution: 3x - y = 2 and 9x - 3y = 6.

Step

Given system of the equations are

3x - y = 2 Equation (1)

9x - 3y = 6 Equation (2)

Formula:

The linear system is

\$ a_1x + b_1y = c_1 \$

\$ a_2x + b_2y = c_2 \$

Hint

By comparing with the linear system, we get

\$ a_1 = 3, b_1 = -1 ,c_1 = 2 , a_2 = 9 , b_2 = - 3 ,c_2 = 6 \$

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) \$

Step

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

Choice.A

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

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

Choice.B

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

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

Choice.C

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

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

Choice.D

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

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

Answer

Option

Sumup

Can you summarize what you’ve understood in the above steps?