Question

3x + 2y = 4
2x - y = 5
This system of equations

has no solution.

has one solution. correct?

is coincident.

Answer 1

Multiply the second equation by 2, and add the two equations together. What do you get?

Answer 2

no solution?

Answer 3

what do you get after doing the steps I described?

Answer 4

i dont know how to do it

Answer 5

\[3x+2y=4\]\[2x-y=5\]Multiply the second equation by 2:
\[2*2x-2*y=2*5\]\[4x-2y=10\]

Add the two equations together:
\[3x+4x + 2y - 2y = 4 + 10\]\[7x + 0y = 14\]Solve that for \(x\)

Answer 6

so would it be coincident?

Answer 7

Please, do as I ask…solve that equation for \(x\).

Answer 8

10x + 4x + 2y

Answer 9

No. \[7x+0y=14\]Solve that for \(x\). You'll get a number.

Answer 10

7?

Answer 11

if x = 7, then
\[7(x) + 0y =14\]\[7(7) + 0 = 14\]\[49 = 14\]Is that true?

Answer 12

yes

Answer 13

49 = 14?!?

Answer 14

In other developments, red is blue, black is white, up is down, left is right, did I miss any? :-)

Answer 15

no. 2

Answer 16

Much better! If \(x=2\), what does \(y=\)?

Answer 17

0

Answer 18

No, we have to go back to one of the previous equations to find \(y\):

\[3x+2y=4\]\[
2x−y=5\]\[3(2) + 2y = 4\]\[6+2y=4\]\[2y=4-6\]\[2y=-2\]\[y=-1\]

or\[2(2)-y=5\]\[4-y=5\]\[-y=1\]\[y=-1\]

So \((2,-1\) is the solution to the system of linear equations. Both of those equations are equations of a line, and \((2,-1)\) is the point at which they int

Answer 19

okkkk
has no solution.

has one solution.

is coincident.

Answer 20

How many solutions did you see? The lines crossed in exactly 1 point, right?

The other cases are:

parallel lines: identical slope, different y-intercepts, don't ever cross, and so there are no solutions. If we go through the solution process, we end up with something like \(0=1\). That's your indication that there are no solutions. The difference in the equation at the end is simply the difference in the y-intercepts of the lines.

coincident lines: identical slope, identical y-intercepts, and so there are infinitely many solutions because any points that appears on one line also appears on the other(s). If you solve the equations as we did, you end up with something like \(0=0\). That's your indication that there are infinitely many solutions.

Answer 21

what? on the 1st paragraph you say theres no solution. and on your second paragraph there are infinitely many solutions.

Answer 22

Those are two different cases, read carefully.

Answer 23

Is no solution the answer

Answer 24

No, we found a solution, didn't we? x=2, y = -1 satisfied both equations!

Answer 25

I then made the mistake of trying to explain all 3 possible answers to you, so that you might recognize them in the future, but apparently only confused you.