Question

Solve x – 2y = 6. 3x – 6y = 18. . A. (2, –2). B. (3, negative three halves) C. No Solutions D. Infinitely Many Solutions

Answer 1

It would be helpful if we write the equations given above in the slope-intercept form which is expressed as:

y = mx + b

We do as follows:

x – 2y = 6
y = x/2 - 3

3x – 6y = 18
y = x/2 - 3

By looking at the equations, we can see that the two equations are the same. Thus, the correct answer is option D. There are infinitely many solutions.

Answer 2

The equations $x - 2y = 6{\text{ and }}3x - 6y = 18$ has infinitely many solutions. Option (D) is correct.

Further explanation:

Consider ${a_1}x + {b_1}y + {c_1}$ and ${a_2}x + {b_2}y + {c_2}.$

If $\dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}}$ then the system of equation has exactly one solution and the system of equations are consistent.

If $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}$ then the system of equation has infinite many solution and the system of equations are consistent.

If $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}}$ then the system of equation has no solution and the system of equations are inconsistent.

Given:

The options are as follows,

A. $\left( {2, - 2} \right)$

B. $\left( {3,{\text{negative three halves}}} \right)$

C. No Solutions

D. Infinitely Many Solutions

Explanation:

The equations are $x - 2y = 6{\text{ and }}3x - 6y = 18.$

${a_1} = 1,{b_1} = - 2{\text{ and }}{c_1} = 6$

${a_2} = 3,{b_2} = - 6{\text{ and }}{c_2} = 18$

The ratio of $\dfrac{{{a_1}}}{{{a_2}}}, \dfrac{{{b_1}}}{{{b_2}}}\, \text{and}\, \dfrac{{{c_1}}}{{{c_2}}}$ can be calculated as follows,

$\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{1}{3}\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{1}{3}\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{1}{3}$

The system of equations has infinite many solutions.

The equations $x - 2y = 6{\text{ and }}3x - 6y = 18$ has infinitely many solutions. Option (D) is correct.

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: Middle School

Subject: Mathematics

Chapter: Linear equation

Keywords: consistent, inconsistent, equations, system of equations, parallel lines, intersecting lines, coincident lines, no solution, many solutions, one solution.