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Answer :
To solve this problem, we're given a system of equations:
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]
We need to determine which of the listed equations can be solved using this system of equations. Here's how we can approach this:
### Step 1: Setting the Equations Equal to Each Other
Since both equations are set equal to [tex]\( y \)[/tex], we can equate them:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
This equation is clearly one of the choices given:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
### Step 2: Individual Equations
Next, consider each equation individually:
- From the first equation: [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex], we can set this equal to zero:
[tex]\[ 3x^3 - 7x^2 + 5 = 0 \][/tex]
- From the second equation: [tex]\( y = 7x^4 + 2x \)[/tex], we can also set this equal to zero:
[tex]\[ 7x^4 + 2x = 0 \][/tex]
Both of these are options as well and correspond to two other choices given:
- [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
- [tex]\( 7x^4 + 2x = 0 \)[/tex]
### Step 3: Combined Polynomial Coefficients
Lastly, let's analyze if there could be a combination or error in initially given equations:
The equation [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex] seems to be a combination of terms from both [tex]\( y \)[/tex] equations. However, it doesn't directly result from the step-by-step process derived from setting [tex]\( y \)[/tex]-components to zero or each other.
### Conclusion
The possible equations that can result from manipulating the given system of equations are:
- [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
- [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
- [tex]\( 7x^4 + 2x = 0 \)[/tex]
Thus, these correspond to the first three choices and can be solved using the system provided.
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]
We need to determine which of the listed equations can be solved using this system of equations. Here's how we can approach this:
### Step 1: Setting the Equations Equal to Each Other
Since both equations are set equal to [tex]\( y \)[/tex], we can equate them:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
This equation is clearly one of the choices given:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
### Step 2: Individual Equations
Next, consider each equation individually:
- From the first equation: [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex], we can set this equal to zero:
[tex]\[ 3x^3 - 7x^2 + 5 = 0 \][/tex]
- From the second equation: [tex]\( y = 7x^4 + 2x \)[/tex], we can also set this equal to zero:
[tex]\[ 7x^4 + 2x = 0 \][/tex]
Both of these are options as well and correspond to two other choices given:
- [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
- [tex]\( 7x^4 + 2x = 0 \)[/tex]
### Step 3: Combined Polynomial Coefficients
Lastly, let's analyze if there could be a combination or error in initially given equations:
The equation [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex] seems to be a combination of terms from both [tex]\( y \)[/tex] equations. However, it doesn't directly result from the step-by-step process derived from setting [tex]\( y \)[/tex]-components to zero or each other.
### Conclusion
The possible equations that can result from manipulating the given system of equations are:
- [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
- [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
- [tex]\( 7x^4 + 2x = 0 \)[/tex]
Thus, these correspond to the first three choices and can be solved using the system provided.
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