1. ## Impossible simultaneous equation?

Hey! I just got this equation in my math homework. I couldnt solve it, and noone else could.
Is this an impossible equation? and if so, why?

Equation:

I: -6x + 3y = 5

II: -16x + 3y = 7

2. ## Re: Impossible simultaneous equation?

There is a solution.

3. ## Re: Impossible simultaneous equation?

Originally Posted by MarkFL2
There is a solution.
Whats the solution? I didnt find any.

4. ## Re: Impossible simultaneous equation?

Try solving both equations for 3y, then equate the results to find x, then using either equation to determine y.

5. ## Re: Impossible simultaneous equation?

Hello, Potrik98!

Hey! I just got this problem in my math homework.
I couldn't solve it, and no one else could.
. . Really? No one?

Is this an impossible equation? .And if so, why?

Equations: .$\displaystyle \begin{array}{cccc}-6x + 3y &=& 5 & [1] \\-16x + 3y &=& 7 & [2] \end{array}$

There are several method for solving system of linear equations.
Which ones are you familiar with?
It seems that you and your classmates have learned NONE of them.

Elimination

We have: .$\displaystyle \begin{array}{cccc}\text{-}6x + 3y &=& 5 & [1] \\\text{-}16x + 3y &=& 7 & [2] \end{array}$

Subtract [1]-[2]: .$\displaystyle 10x \:=\:\text{-}2 \quad\Rightarrow\quad \boxed{x \:=\:\text{-}\tfrac{1}{5}}$

Substitute into [1]: .$\displaystyle \text{-}6\left(\text{-}\tfrac{1}{5}\right) + 3y \:=\:5 \quad\Rightarrow\quad 3y \:=\:\tfrac{19}{5} \quad\Rightarrow\quad \boxed{y \:=\:\tfrac{19}{15}}$

Substitution

Solve [1] for $\displaystyle y\!:\;\text{-}6x + 3y \:=\:5 \quad\Rightarrow\quad 3y \:=\:6x + 5 \quad\Rightarrow\quad y \:=\:2x+\tfrac{5}{3}$

Substitute into [2]: .$\displaystyle \text{-}16x + 3(2x + \tfrac{5}{3}) \:=\:7 \quad\Rightarrow\quad \text{-}16x + 6x + 5 \:=\:7$

. . . . . . . . . . . . . . . $\displaystyle \text{-}10x \:=\:2 \quad\Rightarrow\quad \boxed{x \:=\:\text{-}\tfrac{1}{5}}$

Substitute into [1]: .$\displaystyle \text{-}6\left(\text{-}\tfrac{1}{5}\right) + 3y \:=\:5 \quad\Rightarrow\quad 3y \:=\:\tfrac{19}{5} \quad\Rightarrow\quad \boxed{y \:=\:\tfrac{19}{15}}$

Cramer's Rule

We have: .$\displaystyle \left|\begin{array}{cc|c} \text{-}6 & 3 & 5 \\ \text{-}16 & 3 & 7 \end{array}\right|$

$\displaystyle D \;=\;\begin{vmatrix}-6&3\\-16&3\end{vmatrix} \:=\:(\text{-}18)-(\text{-}48) \:=\:30$

$\displaystyle D_x \:=\:\begin{vmatrix} 5&3\\7&3\end{vmatrix} \:=\:15-21 \:=\:\text{-}6$

. . $\displaystyle x \;=\;\frac{D_x}{D} \;=\;\frac{\text{-}6}{30} \quad\Rightarrow\quad \boxed{x\;=\;\text{-}\tfrac{1}{5}}$

$\displaystyle D_y \:=\:\begin{vmatrix}\text{-}6&5\\\text{-}16&7\end{vmatrix} \:=\:\text{-}42+80 \:=\:38$

. . $\displaystyle y \;=\;\frac{D_y}{D} \;=\;\frac{38}{30} \quad\Rightarrow\quad \boxed{y \:=\:\tfrac{19}{15}}$

6. ## Re: Impossible simultaneous equation?

Thanks for the help, though I misstyped the equation, what I meant was:

I: -6x + 3y = 5

II: -16x + 8y = 7

the + 3y in equation II was really a +8y, though i will try some of the methods you listed

7. ## Re: Impossible simultaneous equation?

Originally Posted by Potrik98
Thanks for the help, though I misstyped the equation, what I meant was:

I: -6x + 3y = 5

II: -16x + 8y = 7

the + 3y in equation II was really a +8y, though i will try some of the methods you listed
To use elimination you need one of the variables to have the same coefficients. So multiply each equation by a number which will give the LCM of 6 and 16 (if you want to eliminate x), or multiply each equation by a number which will give the LCM of 3 and 8 (if you want to eliminate y).

8. ## Re: Impossible simultaneous equation?

Well im still having problems:

[I] -6x + 3y = 5
[II] -16x + 8y = 7

Elimination:

[I] -6x + 3y = 5 | * 8
[II] -16x + 8y = 7 | * -3

[I] -48x + 24y = 40
[II] 48x - 24y = -21

0x + 0y = 19

???
How is this possible?
???

Substutition:

[I] -6x + 3y = 5
[II] -16x + 8y = 7

[I]
-6x + 3y = 5
3y = 5 + 6x | : 3
y = 5/3 + 2x

[II]
-16x + 8y = 7
-16x + 8(5/3 + 2x) = 7
-16x + 40/3 + 16x = 7
-16x + 16x = 7 - 40/3
0x = 7 - 40/3
0x = -19/3

???

This is completely strange. I've tried to slove the equation on several simoultaneous equation solving sites, and my Casio fx-9860 GII calculator, and they all give me the same answer.

Mathemathical error

9. ## Re: Impossible simultaneous equation?

The two equations represent lines in the Cartesian plane...lines with equal slope but different intercepts, hence they are parallel and distinct, and will never cross, so there is no solution.

10. ## Re: Impossible simultaneous equation?

Originally Posted by Potrik98
0x + 0y = 19

???
How is this possible?
???
Exactly. It is not possible, so the two equations do not have solutions. If you ever run into the same situation again, you can safely say that the equations have no simultaneous solutions.

11. ## Re: Impossible simultaneous equation?

Suppose you have two linear equations where none of the coefficients of the variables are zero:

$\displaystyle A_1x+B_1y=C_1$

$\displaystyle A_2x+B_2y=C_2$

A quick and easy check you may perform first to tell you of the nature of the solution is as follows:

If $\displaystyle \frac{A_1}{A_2}=\frac{B_1}{B_2}=\frac{C_1}{C_2}$ then there are an infinite number of solutions, as they are the same equation.

If $\displaystyle \frac{A_1}{A_2}=\frac{B_1}{B_2}\ne\frac{C_1}{C_2}$ then there is no solution, as they are separate, but parallel lines.

If $\displaystyle \frac{A_1}{A_2}\ne\frac{B_1}{B_2}$ then there is a unique solution.