Answer :
To solve this problem, we need to understand what the question is asking. We have the following system of equations:
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]
We need to determine which equation can be derived from this system. The system gives us two expressions for [tex]\( y \)[/tex], which means these two expressions are equal to each other. To find the common equation they satisfy, we should set the expressions equal to each other:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
This equation is the key to solving the problem. Now, let's compare this equation with the options given:
1. [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
2. [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
3. [tex]\( 7x^4 + 2x = 0 \)[/tex]
4. [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex]
The equation we derived, [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex], directly matches option 2.
Therefore, the correct answer is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]
We need to determine which equation can be derived from this system. The system gives us two expressions for [tex]\( y \)[/tex], which means these two expressions are equal to each other. To find the common equation they satisfy, we should set the expressions equal to each other:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
This equation is the key to solving the problem. Now, let's compare this equation with the options given:
1. [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
2. [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
3. [tex]\( 7x^4 + 2x = 0 \)[/tex]
4. [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex]
The equation we derived, [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex], directly matches option 2.
Therefore, the correct answer is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
