Finding the Point of Intersection: Is (-2, -2.5) the Solution?

[nicholar1]

Hi,

I am looking for some help on how you find the point of intersection of the following two lines:

4y = x - 8
2y = 3x + 1

Thanks for any help.

/Nichola

 

[SuperSonic4]

Hi,

I am looking for some help on how you find the point of intersection of the following two lines:

4y = x - 8
2y = 3x + 1

Thanks for any help.

/Nichola

Hi Nichola

The intersection will be at the point where the equations are equal. Are you familiar with solving simultaneous equations either by substitution or elimination?

Spoiler

Multiply the second equation by 2:

\(\displaystyle 4y = 6x+2\)

You can then subtract this equation from the first one to eliminate y

 

[nicholar1]

Hi Nichola

The intersection will be at the point where the equations are equal. Are you familiar with solving simultaneous equations either by substitution or elimination?

Spoiler

Multiply the second equation by 2:

\(\displaystyle 4y = 6x+2\)

You can then subtract this equation from the first one to eliminate y

Hi,

No I have never done simultaneous equations before, that's where I'm getting confused, how they work together.

 

[I like Serena]

Hi,

I am looking for some help on how you find the point of intersection of the following two lines:

4y = x - 8
2y = 3x + 1

Thanks for any help.

/Nichola

Hi,

No I have never done simultaneous equations before, that's where I'm getting confused, how they work together.

Welcome to MHB Nichola! :)

We can multiply the second equation by 2 as Supersonic suggested.
That is, we start with:
$$2y = 3x + 1$$
and we end up with
$$4y = 6x + 2$$

Now we can see that according to the first equation $4y=x-8$ and according to the rewritten second equation we also have that $4y=6x+2$.
Since both are equal to $4y$ it must be that $x-8$ is equal to $6x+2$.

How would you solve the equation $x-8 = 6x+2$?

 

[nicholar1]

Welcome to MHB Nichola! :)

We can multiply the second equation by 2 as Supersonic suggested.
That is, we start with:
$$2y = 3x + 1$$
and we end up with
$$4y = 6x + 2$$

Now we can see that according to the first equation $4y=x-8$ and according to the rewritten second equation we also have that $4y=6x+2$.
Since both are equal to $4y$ it must be that $x-8$ is equal to $6x+2$.

How would you solve the equation $x-8 = 6x+2$?

So to solve the equation $x-8 = 6x+2$ we:

Add 8 to both sides: $x = 6x+10$
Subtract 6x from both sides: $-5x = 10$
Divide both sides by -5: $x = -2$

So, substituting x = -2 into the equations give:

$4y = -2 - 8 = -10$

$2y = 3 x (-2) + 1 = -5$

So dividing -10 by 4 or -5 by 2 gives -2.5.

So does that mean the point of intersection is (-2, -2.5)?

Thanks for the help guys :)

/Nichola

 

[I like Serena]

So to solve the equation $x-8 = 6x+2$ we:

Add 8 to both sides: $x = 6x+10$
Subtract 6x from both sides: $-5x = 10$
Divide both sides by -5: $x = -2$

So, substituting x = -2 into the equations give:

$4y = -2 - 8 = -10$

$2y = 3 x (-2) + 1 = -5$

So dividing -10 by 4 or -5 by 2 gives -2.5.

So does that mean the point of intersection is (-2, -2.5)?

Thanks for the help guys :)

/Nichola

Let's verify...

Substiting (-2, -2.5) in:
\begin{aligned}4y &= x - 8 \\
2y &= 3x + 1
\end{aligned}
gives:
\begin{aligned}4 \cdot -2.5 &= -2 -8 \\
2 \cdot -2.5 &= 3 \cdot -2 + 1
\end{aligned}
simplifying:
\begin{aligned}-10 &= -10 \\
-5 &= -5
\end{aligned}

We have a match, so this is the correct solution! (Happy)