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Answer :
To determine which equation can be solved by using the given system of equations, let's follow these steps:
### Given System of Equations:
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]
### Method:
To find a compatible equation that can be used with this system, we set both expressions for [tex]\( y \)[/tex] equal to each other because they represent the same [tex]\( y \)[/tex] value in the system of equations.
### Step-by-Step Solution:
1. Set the Equations Equal:
[tex]\[
3x^3 - 7x^2 + 5 = 7x^4 + 2x
\][/tex]
2. Rearrange the Equation:
- Move all terms to one side to set the equation to zero:
[tex]\[
0 = 7x^4 + 2x - 3x^3 + 7x^2 - 5
\][/tex]
3. Identify the Correct Form:
- From the given choices, the equation that matches what we formulated is:
[tex]\[
3x^3 - 7x^2 + 5 = 7x^4 + 2x
\][/tex]
This equation, [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex], correctly represents the problem of equating both equations for [tex]\( y \)[/tex] as described in the system, and is the appropriate equation from the given list.
### Given System of Equations:
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]
### Method:
To find a compatible equation that can be used with this system, we set both expressions for [tex]\( y \)[/tex] equal to each other because they represent the same [tex]\( y \)[/tex] value in the system of equations.
### Step-by-Step Solution:
1. Set the Equations Equal:
[tex]\[
3x^3 - 7x^2 + 5 = 7x^4 + 2x
\][/tex]
2. Rearrange the Equation:
- Move all terms to one side to set the equation to zero:
[tex]\[
0 = 7x^4 + 2x - 3x^3 + 7x^2 - 5
\][/tex]
3. Identify the Correct Form:
- From the given choices, the equation that matches what we formulated is:
[tex]\[
3x^3 - 7x^2 + 5 = 7x^4 + 2x
\][/tex]
This equation, [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex], correctly represents the problem of equating both equations for [tex]\( y \)[/tex] as described in the system, and is the appropriate equation from the given list.
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