Proof involving two linear equations

[RM86Z]

TL;DR

Given $a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0$, if $(x_1,y_1)$ and $(x_2,y_2)$ are pairs of real numbers satisfying:

$ax_1 + by_1 = e, cx_1 + dy_1 =f$
$ax_2 + by_2 = e, cx_2 + dy_2 = f$

then $(x_1,y_1) = (x_2,y_2).$

Given $a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0$, if $(x_1,y_1)$ and $(x_2,y_2)$ are pairs of real numbers satisfying:

$ax_1 + by_1 = e, cx_1 + dy_1 =f$
$ax_2 + by_2 = e, cx_2 + dy_2 = f$

then $(x_1,y_1) = (x_2,y_2).$

Here is my attempt at a proof, I have gotten stuck:
1)
$ax_1 + by_1 = ax_2 + by_2$
$cx_1 + dy_1 = cx_2 + dy_2$

2)
$a(x_1 - x_2) = b(y_2 - y_1)$
$c(x_1 - x_2) = d(y_2 - y_1)$

3)
$\frac{a}{b} = \frac{y_2 - y_1}{x_1 - x_2}$
$\frac{c}{d} = \frac{y_2 - y_1}{x_1 - x_2}$

4)
$\frac{a}{b} = \frac{c}{d}$

5)
$ad = bc$

6)
$ad - bc = 0$ but this is false as the hypothesis states that $ad - bc \neq 0.$

I am stuck now and am not sure how to proceed.

 

[PeroK]

Isn't that essentially a proof? What assumptions did you make in order to get $ad = bc$?

Note: have you studied matrices yet?

 

[RM86Z]

I don't think it is a proof as I need to prove that $(x_1,y_1)$ and $(x_2,y_2)$ are equal, or in other words $x_1 - x_2 = 0$ and $y_1 - y_2 = 0.$ I cannot see how to get there from the result I have ended with?

I have done some linear algebra a long time ago but I am very rusty on everything and am just trying to learn some proofs as that's something I haven't done before. I can't remember all the details of the determinant at this stage...

 

[PeroK]

3)
$\frac{a}{b} = \frac{y_2 - y_1}{x_1 - x_2}$
$\frac{c}{d} = \frac{y_2 - y_1}{x_1 - x_2}$

4)
$\frac{a}{b} = \frac{c}{d}$

What are you assuming here?

 

[RM86Z]

Sorry I know there is a connection here I'm just not getting it at this stage. By the properties of the determinant the result $ad - bc = 0$ tells me that the two rows of the linear system are the same so $a = b$ and $c = d.$

 

[PeroK]

Sorry I know there is a connection here I'm just not getting it at this stage. By the properties of the determinant the result $ad - bc = 0$ tells me that the two rows of the linear system are the same so $a = b$ and $c = d.$

When is this not valid?

3)
$\frac{a}{b} = \frac{y_2 - y_1}{x_1 - x_2}$
$\frac{c}{d} = \frac{y_2 - y_1}{x_1 - x_2}$

 

[RM86Z]

When $x_1 - x_2 = 0$ and actually now that I think about it it wasn't defined in the hypothesis that $b,d \neq 0$ as that certainly isn't valid either?

EDIT: I suppose the last bit about $a,b \neq 0$ doesn't make sense as it depends on the ordering of the algebra. In other words instead of $\frac{a}{b}$ we could have $\frac{b}{a}$...

 

[PeroK]

When $x_1 - x_2 = 0$ and actually now that I think about it it wasn't defined in the hypothesis that $b,d \neq 0$ as that certainly isn't valid either?

You tacitly assumed that $x_1 \ne x_2$ and reached a contradiction - in terms of $ad = bc$. You've effectively proved, therefore, that $x_1 = x_2$, as that is the only way around the contradiction.

You always need to be careful that the denominator on any expression in not zero. Technically, you should have done that in going from step 2) to step 3)

2)
$a(x_1 - x_2) = b(y_2 - y_1)$
$c(x_1 - x_2) = d(y_2 - y_1)$

3)
$\frac{a}{b} = \frac{y_2 - y_1}{x_1 - x_2}$
$\frac{c}{d} = \frac{y_2 - y_1}{x_1 - x_2}$

You should have indicated here that you were assuming that $x_1 - x_2 \ne 0$. Otherwise, that is not valid mathematically.

 

[PeroK]

PS In fact, it's not valid if $b = 0$ or $d = 0$ either!

 

[RM86Z]

Ah brilliant that makes sense now thank you so much PeroK! That was a very slight detail that I wasn't seeing. I appreciate the help.

 

[fresh_42]

Your equations can be written as
$$
\begin{bmatrix}a&b\\c&d\end{bmatrix}\cdot \begin{bmatrix}x_1\\y_1\end{bmatrix}=\begin{bmatrix}e\\f \end{bmatrix}\text{ and }\begin{bmatrix}a&b\\c&d\end{bmatrix}\cdot \begin{bmatrix}x_2\\y_2\end{bmatrix}=\begin{bmatrix}e\\f \end{bmatrix}
$$
From there you can use
$$
\begin{bmatrix}d&-b\\-c&a \end{bmatrix}\cdot\begin{bmatrix}a&b\\c&d\end{bmatrix}=(ad-bc)\cdot\begin{bmatrix}1&0\\0&1\end{bmatrix}
$$

 

[mathman]

Summary:: Given $a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0$, if $(x_1,y_1)$ and $(x_2,y_2)$ are pairs of real numbers satisfying:

$ax_1 + by_1 = e, cx_1 + dy_1 =f$
$ax_2 + by_2 = e, cx_2 + dy_2 = f$

then $(x_1,y_1) = (x_2,y_2).$

Given $a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0$, if $(x_1,y_1)$ and $(x_2,y_2)$ are pairs of real numbers satisfying:

$ax_1 + by_1 = e, cx_1 + dy_1 =f$
$ax_2 + by_2 = e, cx_2 + dy_2 = f$

then $(x_1,y_1) = (x_2,y_2).$

Here is my attempt at a proof, I have gotten stuck:
1)
$ax_1 + by_1 = ax_2 + by_2$
$cx_1 + dy_1 = cx_2 + dy_2$

2)
$a(x_1 - x_2) = b(y_2 - y_1)$
$c(x_1 - x_2) = d(y_2 - y_1)$

3)
$\frac{a}{b} = \frac{y_2 - y_1}{x_1 - x_2}$
$\frac{c}{d} = \frac{y_2 - y_1}{x_1 - x_2}$

4)
$\frac{a}{b} = \frac{c}{d}$

5)
$ad = bc$

6)
$ad - bc = 0$ but this is false as the hypothesis states that $ad - bc \neq 0.$

I am stuck now and am not sure how to proceed.

Each $(x_k,y_k)$ pair will have a unique solution. Since all constants are the same for both, the pairs must be equal.

 

[fresh_42]

Each $(x_k,y_k)$ pair will have a unique solution.

This is the statement that has to be shown. It is not automatically obvious.

Since all constants are the same for both, the pairs must be equal.

This is wrong. E.g. $a=b=c=d=e=f=1.$

 

[mathman]

a=a, b=b, etc. is what I meant.
I didn't think it was necessary to prove what is well known. Two linear equations in two unknowns has either a unique solution or none. Basic point: both pairs of equations use the same (a,b, etc.) they will have the same solution. The unknowns are dummies, so it doesn't matter what pair of letters are used, the answer will always be the same for a given set of knowns.

 

[fresh_42]

Two linear equations in two unknowns has either a unique solution or none.

Or infinitely many.

 

[PeroK]

I didn't think it was necessary to prove what is well known.

It's not well known to someone just learning the subject!

 

[fresh_42]

Summary:: Given $a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0$, if $(x_1,y_1)$ and $(x_2,y_2)$ are pairs of real numbers satisfying:

$ax_1 + by_1 = e, cx_1 + dy_1 =f$
$ax_2 + by_2 = e, cx_2 + dy_2 = f$

then $(x_1,y_1) = (x_2,y_2).$

Given $a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0$, if $(x_1,y_1)$ and $(x_2,y_2)$ are pairs of real numbers satisfying:

$ax_1 + by_1 = e, cx_1 + dy_1 =f$
$ax_2 + by_2 = e, cx_2 + dy_2 = f$

then $(x_1,y_1) = (x_2,y_2).$

You can consider $ax + by = e$ as the equation for a straight and $cx + dy =f$ as the equation of a second straight. Both straights are in the $(x,y)$ coordinate plane. Now, either those two lines are a) equal, b) parallel, or c) intersect in one point.

$ad-bc\neq 0$ rules out the first two cases, but this needs some considerations to see.

 

[Office_Shredder]

You can consider $ax + by = e$ as the equation for a straight and $cx + dy =f$ as the equation of a second straight. Both straights are in the $(x,y)$ coordinate plane. Now, either those two lines are a) equal, b) parallel, or c) intersect in one point.

$ad-bc\neq 0$ rules out the first two cases, but this needs some considerations to see.

Somehow I've never thought of it this way. Genius.