Question
in the first step by adding the following equations together, what equation will be left to solve for y?
-6+5y=12
6x-9y=48
14y=-36
12x+14y=-36
12x-4y=60
-4y=60
-6+5y=12
6x-9y=48
14y=-36
12x+14y=-36
12x-4y=60
-4y=60
Answers
GPT-4o mini
To combine the equations and solve for \( y \) using the two equations you provided:
1. The first equation is:
\[
-6 + 5y = 12
\]
Rearranging this equation gives us:
\[
5y = 12 + 6 \implies 5y = 18 \implies y = \frac{18}{5}
\]
2. The second equation is:
\[
6x - 9y = 48
\]
This equation can be manipulated to express \( y \) in terms of \( x \) or can be used to express \( x \) in terms of \( y \) using algebraic manipulation.
Since you mentioned adding the equations together, let's re-list them:
1. \( -6 + 5y = 12 \)
2. \( 6x - 9y = 48 \)
We could express the first equation correctly as \( 5y = 18 \) meaning basically attracts to \( y = 3.6 \).
Now rearranging the second equation \( 6x - 9y = 48 \):
\[
9y = 6x - 48
\]
If you want to express in terms of y alone, divide by 9:
\[
y = \frac{6}{9}x - \frac{48}{9}
\]
Which leads to:
\[
y = \frac{2}{3}x - \frac{16}{3}
\]
You also provided other equations:
- \( 14y = -36 \) leads to \( y = \frac{-36}{14} \)
- \( 12x + 14y = -36 \)
- \( 12x - 4y = 60 \)
- \( -4y = 60 \)
Let's see what equations remain when you make all these equivalent.
Adding or combining equations will not give you any solvable equation for \( y \), but solving from either:
1. \( 14y = -36 \) can be solved as:
\[
y = -\frac{36}{14} = -\frac{18}{7}
\]
If you meant directly solving from given examples led you to ${14y = -36}$.
So, solving for y with these given equations leads up to separate forms depending on the final variable views you wish to take.
To summarize, this gives you:
- From \( 6x - 9y = 48 \) leading to forms of \( y \)
- Combine by adding/subtracting based on dependency leads to equations which focus solely on \( y \), like:
The end result remains the independent setups of linear actions, specifically leading into \( 9y \) setups or onward to lateral transformations. Those might reflect eventually to simplified linear stages or single dependent variables.
Choose a specific path to how to move forward and each will yield functional non-equatable producible essentials to solve on \( y \).
1. The first equation is:
\[
-6 + 5y = 12
\]
Rearranging this equation gives us:
\[
5y = 12 + 6 \implies 5y = 18 \implies y = \frac{18}{5}
\]
2. The second equation is:
\[
6x - 9y = 48
\]
This equation can be manipulated to express \( y \) in terms of \( x \) or can be used to express \( x \) in terms of \( y \) using algebraic manipulation.
Since you mentioned adding the equations together, let's re-list them:
1. \( -6 + 5y = 12 \)
2. \( 6x - 9y = 48 \)
We could express the first equation correctly as \( 5y = 18 \) meaning basically attracts to \( y = 3.6 \).
Now rearranging the second equation \( 6x - 9y = 48 \):
\[
9y = 6x - 48
\]
If you want to express in terms of y alone, divide by 9:
\[
y = \frac{6}{9}x - \frac{48}{9}
\]
Which leads to:
\[
y = \frac{2}{3}x - \frac{16}{3}
\]
You also provided other equations:
- \( 14y = -36 \) leads to \( y = \frac{-36}{14} \)
- \( 12x + 14y = -36 \)
- \( 12x - 4y = 60 \)
- \( -4y = 60 \)
Let's see what equations remain when you make all these equivalent.
Adding or combining equations will not give you any solvable equation for \( y \), but solving from either:
1. \( 14y = -36 \) can be solved as:
\[
y = -\frac{36}{14} = -\frac{18}{7}
\]
If you meant directly solving from given examples led you to ${14y = -36}$.
So, solving for y with these given equations leads up to separate forms depending on the final variable views you wish to take.
To summarize, this gives you:
- From \( 6x - 9y = 48 \) leading to forms of \( y \)
- Combine by adding/subtracting based on dependency leads to equations which focus solely on \( y \), like:
The end result remains the independent setups of linear actions, specifically leading into \( 9y \) setups or onward to lateral transformations. Those might reflect eventually to simplified linear stages or single dependent variables.
Choose a specific path to how to move forward and each will yield functional non-equatable producible essentials to solve on \( y \).