Question

1.  Lisa is working with the system of equations x + 2y = 7 and 2x ‒ 5y = 5. She multiplies the first equation by 2 and then subtracts the second equation to find 9y = 9, telling her that y = 1. Lisa then finds that x = 5. Thinking about this procedure, Lisa wonders:
There are lots of ways I could go about solving this problem. I could add 5 times the first equation and twice the second or I could multiply the first equation by ‒2 and add the second. I seem to find that there is only one solution to the two equations but I wonder if I will get the same solution if I use a different method?

What is the answer to Lisa’s question? Explain.

2.  Does the answer to the first question change if we have a system of two equations in two unknowns with no solutions? What if there are infinitely many solutions?

Answer 1

All methods are supposed to be correct in most if not all situations and it should not matter if you use a different method given the options

Answer 2

Answer:

1. Yes

2. Yes

Step-by-step explanation:

1. Lets test Lisa's different solution. Lets times 5 to the first equation:

$5\cdot{x}+10\cdot{y}=35$

and 2 times the second equation:

$4\cdot{x}-10\cdot{y}=10$

lets add the two equations:

$9\cdot{x}=45$

$x=5$

The second method is multiplying the first equation by -2:

$-2\cdot{x}-4\cdot{y}=-14$

and add the second equation:

$-9\cdot{y}=-9$

$y=1[tex]

Substitute into equation 1:

[tex]x+2\cdot{1}=7$

$x=5$

The answer to Lisa's question is yes she wull get the same solution if she uses a different method.

2. Yes, The answer would change if the same amout of x and y values are the same and therefore we cannot solve for x and y. If there was infinitely many solutions we would have quadratic equations.