Answer :
Sure! Let's solve this step-by-step.
We are given the following system of equations:
[tex]\[
\left\{
\begin{array}{l}
y = 3x^3 - 7x^2 + 5 \\
y = 7x^4 + 2x
\end{array}
\right.
\][/tex]
We need to determine which given equation can be solved by using this system. To do this, we aim to find a relationship between the two equations that eliminates [tex]\(y\)[/tex].
First, observe that both equations are already solved for [tex]\(y\)[/tex]:
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]
Since both expressions equal [tex]\(y\)[/tex], we can set them equal to each other:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
Now let's simplify this equation by moving all terms to one side of the equation (usually to the left side) to get everything into a single equation set to zero:
[tex]\[ 3x^3 - 7x^2 + 5 - (7x^4 + 2x) = 0 \][/tex]
Combine like terms:
[tex]\[ -7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0 \][/tex]
This is a polynomial equation that can be solved for [tex]\(x\)[/tex].
Now let’s look at the provided options to see which one matches our simplified equation:
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 correct answer is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
This equation is the same as the one derived by setting both expressions for [tex]\(y\)[/tex] equal to each other.
So the answer is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
We are given the following system of equations:
[tex]\[
\left\{
\begin{array}{l}
y = 3x^3 - 7x^2 + 5 \\
y = 7x^4 + 2x
\end{array}
\right.
\][/tex]
We need to determine which given equation can be solved by using this system. To do this, we aim to find a relationship between the two equations that eliminates [tex]\(y\)[/tex].
First, observe that both equations are already solved for [tex]\(y\)[/tex]:
1. [tex]\( y = 3x^3 - 7x^2 + 5 \)[/tex]
2. [tex]\( y = 7x^4 + 2x \)[/tex]
Since both expressions equal [tex]\(y\)[/tex], we can set them equal to each other:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
Now let's simplify this equation by moving all terms to one side of the equation (usually to the left side) to get everything into a single equation set to zero:
[tex]\[ 3x^3 - 7x^2 + 5 - (7x^4 + 2x) = 0 \][/tex]
Combine like terms:
[tex]\[ -7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0 \][/tex]
This is a polynomial equation that can be solved for [tex]\(x\)[/tex].
Now let’s look at the provided options to see which one matches our simplified equation:
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 correct answer is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
This equation is the same as the one derived by setting both expressions for [tex]\(y\)[/tex] equal to each other.
So the answer is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
