Answer :
- Set the two equations equal to each other: $3x^3 - 7x^2 + 5 = 7x^4 + 2x$.
- Rearrange the equation to get all terms on one side: $7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0$.
- Observe that the equation $3 x^3-7 x^2+5=7 x^4+2 x$ is equivalent to the derived equation.
- Conclude that the equation that can be solved using the system of equations is $3 x^3-7 x^2+5=7 x^4+2 x$, which is equivalent to $7 x^4-3 x^3+7 x^2+2 x-5 = 0$.
$\boxed{3 x^3-7 x^2+5=7 x^4+2 x}$
### Explanation
1. Understanding the Problem
We are given a system of two equations: $y=3 x^3-7 x^2+5$ and $y=7 x^4+2 x$. The question asks us to identify which of the provided equations can be derived from this system.
2. Equating the Expressions for y
To find the equation that can be solved using the given system, we can set the two expressions for $y$ equal to each other:$$3x^3 - 7x^2 + 5 = 7x^4 + 2x$$
3. Rearranging the Equation
Now, let's rearrange the equation to match one of the given options. We can rewrite the equation as:$$7x^4 + 2x = 3x^3 - 7x^2 + 5$$or$$7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0$$Multiplying by -1, we get $$-7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0$$However, none of these directly match the options provided. Let's rearrange the terms to have all terms on one side:$$0 = 7x^4 - 3x^3 + 7x^2 + 2x - 5$$Multiplying by -1 gives:$$0 = -7x^4 + 3x^3 - 7x^2 - 2x + 5$$Adding $7x^4$ to both sides of the original equation $3x^3 - 7x^2 + 5 = 7x^4 + 2x$, we get:$$7x^4 + 3x^3 - 7x^2 + 2x - 5 = 0$$
4. Identifying the Correct Equation
Comparing the derived equation $7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0$ with the given options, we see that the equation $3 x^3-7 x^2+5=7 x^4+2 x$ is equivalent to $7x^4 + 2x = 3x^3 - 7x^2 + 5$, which can be rearranged to $7x^4 - 3x^3 + 2x + 7x^2 - 5 = 0$. Thus, the equation $3 x^3-7 x^2+5=7 x^4+2 x$ can be solved using the given system of equations.
### Examples
In engineering, when designing systems with multiple components, you often encounter scenarios where two different models predict the same outcome (like the 'y' in our equations). Finding the conditions where these models agree (solving for 'x') is crucial for ensuring the system's stability and reliability. This could apply to anything from electrical circuits to mechanical structures, where different analysis methods must converge for a safe and efficient design.
- Rearrange the equation to get all terms on one side: $7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0$.
- Observe that the equation $3 x^3-7 x^2+5=7 x^4+2 x$ is equivalent to the derived equation.
- Conclude that the equation that can be solved using the system of equations is $3 x^3-7 x^2+5=7 x^4+2 x$, which is equivalent to $7 x^4-3 x^3+7 x^2+2 x-5 = 0$.
$\boxed{3 x^3-7 x^2+5=7 x^4+2 x}$
### Explanation
1. Understanding the Problem
We are given a system of two equations: $y=3 x^3-7 x^2+5$ and $y=7 x^4+2 x$. The question asks us to identify which of the provided equations can be derived from this system.
2. Equating the Expressions for y
To find the equation that can be solved using the given system, we can set the two expressions for $y$ equal to each other:$$3x^3 - 7x^2 + 5 = 7x^4 + 2x$$
3. Rearranging the Equation
Now, let's rearrange the equation to match one of the given options. We can rewrite the equation as:$$7x^4 + 2x = 3x^3 - 7x^2 + 5$$or$$7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0$$Multiplying by -1, we get $$-7x^4 + 3x^3 - 7x^2 - 2x + 5 = 0$$However, none of these directly match the options provided. Let's rearrange the terms to have all terms on one side:$$0 = 7x^4 - 3x^3 + 7x^2 + 2x - 5$$Multiplying by -1 gives:$$0 = -7x^4 + 3x^3 - 7x^2 - 2x + 5$$Adding $7x^4$ to both sides of the original equation $3x^3 - 7x^2 + 5 = 7x^4 + 2x$, we get:$$7x^4 + 3x^3 - 7x^2 + 2x - 5 = 0$$
4. Identifying the Correct Equation
Comparing the derived equation $7x^4 - 3x^3 + 7x^2 + 2x - 5 = 0$ with the given options, we see that the equation $3 x^3-7 x^2+5=7 x^4+2 x$ is equivalent to $7x^4 + 2x = 3x^3 - 7x^2 + 5$, which can be rearranged to $7x^4 - 3x^3 + 2x + 7x^2 - 5 = 0$. Thus, the equation $3 x^3-7 x^2+5=7 x^4+2 x$ can be solved using the given system of equations.
### Examples
In engineering, when designing systems with multiple components, you often encounter scenarios where two different models predict the same outcome (like the 'y' in our equations). Finding the conditions where these models agree (solving for 'x') is crucial for ensuring the system's stability and reliability. This could apply to anything from electrical circuits to mechanical structures, where different analysis methods must converge for a safe and efficient design.
