Question

To solve the system of linear equations 3x-2y=4 and 9x-6y=12 by using the linear combination method, Henry decided that he should first multiply the first equation by –3 and then add the two equations together to eliminate the x-terms. When he did so, he also eliminated the y-terms and got the equation 0 = 0, so he thought that the system of equations must have an infinite number of solutions. To check his answer, he graphed the equations 3x-2y=4 and 9x-6y=12 with his graphing calculator, but he could only see one line. Why is this?

Answer 1

Henry could only see one line because when simplified, both of those equations give you the same expression: y = 3/2x - 2

Answer 2

Answer:

Both lines have same slope so the graph of both equation will be same and hence it will overlap each other that is why the Henry  could only see one line.

Step-by-step explanation:

Given  : A system of linear equation 3x - 2y = 4 and 9x - 6y = 12  

We have to show whey when Henry  graphed the equations 3x-2y=4 and 9x-6y=12 he could only see one line.

Consider the given system of linear equation

3x - 2y = 4 ................(1)

9x - 6y = 12   ..................(2)

Since,

Equation (2) is multiple of equation (1),

3 × ( 3x - 2y = 4) = 9x - 6y = 12  

Also the slope of given equations are same

For equation (1),

Differentiate with respect to x, we get,

$3-2\frac{dy}{dx}=0\\\\ \frac{dy}{dx}=\frac{3}{2}$

Also for equation (2),

Differentiate with respect to x, we get,

$9-6\frac{dy}{dx}=0\\\\ \frac{dy}{dx}=\frac{9}{6}=\frac{3}{2}$

Since, both lines have same slope so the graph of both equation will be same and hence it will overlap each other.

That is why the Henry  could only see one line.