Answer :
Certainly! Let's analyze the given system of equations and determine which single equation can be derived from it.
The system of equations is:
[tex]\[ \begin{cases} y = 3x^5 - 5x^3 + 2x^2 - 10x + 4 \\ y = 4x^4 + 6x^3 - 11 \end{cases} \][/tex]
To find the equation that can be solved using this system, we will set the two expressions for [tex]\( y \)[/tex] equal to each other since they both equal [tex]\( y \)[/tex]:
[tex]\[ 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11 \][/tex]
This is the equation derived from the system of equations. Now, let's check which option matches this derived equation:
1. [tex]\( 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 0 \)[/tex]
This is not the correct equation because the right-hand side should not be zero.
2. [tex]\( 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11 \)[/tex]
This is the correct equation derived from setting the two expressions for [tex]\( y \)[/tex] equal to each other.
3. [tex]\( 3x^5 + 4x^4 + x^3 + 2x^2 - 10x - 7 = 0 \)[/tex]
This is not correct because the terms and constants do not match the derived equation.
4. [tex]\( 4x^4 + 6x^3 - 11 = 0 \)[/tex]
This is not correct because it is not derived by setting the two equations for [tex]\( y \)[/tex] equal to each other.
Therefore, the correct answer is:
[tex]\[ 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11 \][/tex]
The system of equations is:
[tex]\[ \begin{cases} y = 3x^5 - 5x^3 + 2x^2 - 10x + 4 \\ y = 4x^4 + 6x^3 - 11 \end{cases} \][/tex]
To find the equation that can be solved using this system, we will set the two expressions for [tex]\( y \)[/tex] equal to each other since they both equal [tex]\( y \)[/tex]:
[tex]\[ 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11 \][/tex]
This is the equation derived from the system of equations. Now, let's check which option matches this derived equation:
1. [tex]\( 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 0 \)[/tex]
This is not the correct equation because the right-hand side should not be zero.
2. [tex]\( 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11 \)[/tex]
This is the correct equation derived from setting the two expressions for [tex]\( y \)[/tex] equal to each other.
3. [tex]\( 3x^5 + 4x^4 + x^3 + 2x^2 - 10x - 7 = 0 \)[/tex]
This is not correct because the terms and constants do not match the derived equation.
4. [tex]\( 4x^4 + 6x^3 - 11 = 0 \)[/tex]
This is not correct because it is not derived by setting the two equations for [tex]\( y \)[/tex] equal to each other.
Therefore, the correct answer is:
[tex]\[ 3x^5 - 5x^3 + 2x^2 - 10x + 4 = 4x^4 + 6x^3 - 11 \][/tex]
