How to quickly solve systems of linear equations?

[JUWON]

How do I quickly solve questions that ask for how many solutions a problem has?

For example,

*x + 5 = *x - 5

How many solutions does this have?

1, 2, no solution, infinite

Obviously, the question is a little more complicated than this. But when I am taking a test with little time, do I have to solve the whole problem by hand? Or are there easier methods that I should know.

 

[ksdhart2]

I'm guessing that, in this context, "*" does not have its typical meaning of multiplication. However, I am at a loss as what it does mean. Please reply with the meaning of his unusual notation as well as any and all work you've done on this problem, even the parts you know for sure are wrong. Thank you.

 

[JUWON]

I'm guessing that, in this context, "*" does not have its typical meaning of multiplication. However, I am at a loss as what it does mean. Please reply with the meaning of his unusual notation as well as any and all work you've done on this problem, even the parts you know for sure are wrong. Thank you.

Hi, sorry for being confusing. What I meant would be something like this. 5x - 3 = 5x - 3

This would be infinite solutions if I did 5x - 5x -> - 3 = - 3. That is relatively simple but there are some other questions such as.

How many ordered pairs (x, y) satisfy the system of equations shown above?

x = 2y + 5
y = (2x - 3)(x + 9)

A) 0
B) 1
C) 2
D) Infinitely many

I am studying for the SAT and this is one of the questions I stumbled upon in my practice book. how would I solve this?

 

[Deleted member 4993]

Hi, sorry for being confusing. What I meant would be something like this. 5x - 3 = 5x - 3

This would be infinite solutions if I did 5x - 5x -> - 3 = - 3. That is relatively simple but there are some other questions such as.

How many ordered pairs (x, y) satisfy the system of equations shown above?

x = 2y + 5
y = (2x - 3)(x + 9)

A) 0
B) 1
C) 2
D) Infinitely many

I am studying for the SAT and this is one of the questions I stumbled upon in my practice book. how would I solve this?

To answer this question, you would need to solve several steps toward the solution.

 

[HallsofIvy]

An obvious first step- since x= 2y+5, y= (2x- 3)(x+ 9)= (2(2y+5)- 3)(2y+ 5+ 9)= (4y+ 7)(2y+ 14).

That gives a quadratic equation for y. Once you know y, x is given by x= 2y+ 5.