Answer :
Sure, let's solve the problem step-by-step.
We are given a system of two equations:
```
1) y = 3x^3 - 7x^2 + 5
2) y = 7x^4 + 2x
```
We need to find out which among the given equations can be solved using this system:
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]
Let's analyze each option:
### Option 1: [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
For this to hold true, we substitute [tex]\(3x^3 - 7x^2 + 5\)[/tex] directly into the first equation. This means we are setting [tex]\(y = 0\)[/tex]. The other equation would need to also equal to [tex]\(0\)[/tex] under the same conditions, but it doesn't necessarily hold for all solutions. Therefore, it doesn't fully represent the system.
### Option 2: [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
This option equates the two given equations. By substituting [tex]\(y = 3x^3 - 7x^2 + 5\)[/tex] and [tex]\(y = 7x^4 + 2x\)[/tex], we get:
```
3x^3 - 7x^2 + 5 = 7x^4 + 2x
```
This equation directly matches, representing both sides of the system being equal. Therefore, this equation can be solved using the system.
### Option 3: [tex]\( 7x^4 + 2x = 0 \)[/tex]
For this to hold true, we substitute [tex]\(7x^4 + 2x\)[/tex] directly into the second equation. As a result, y = 0 must also hold true. Similar to option 1, it doesn't necessarily represent all solutions of the system.
### Option 4: [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex]
In this option, all terms from both equations are combined into a single equation. However, it doesn’t isolate [tex]\(y\)[/tex] on one side and doesn’t directly correspond to an equation derived from the system. Hence, it does not fully represent the system either.
Given this step-by-step analysis, the equation that can be solved by using the system of equations is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
So the correct answer is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
We are given a system of two equations:
```
1) y = 3x^3 - 7x^2 + 5
2) y = 7x^4 + 2x
```
We need to find out which among the given equations can be solved using this system:
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]
Let's analyze each option:
### Option 1: [tex]\( 3x^3 - 7x^2 + 5 = 0 \)[/tex]
For this to hold true, we substitute [tex]\(3x^3 - 7x^2 + 5\)[/tex] directly into the first equation. This means we are setting [tex]\(y = 0\)[/tex]. The other equation would need to also equal to [tex]\(0\)[/tex] under the same conditions, but it doesn't necessarily hold for all solutions. Therefore, it doesn't fully represent the system.
### Option 2: [tex]\( 3x^3 - 7x^2 + 5 = 7x^4 + 2x \)[/tex]
This option equates the two given equations. By substituting [tex]\(y = 3x^3 - 7x^2 + 5\)[/tex] and [tex]\(y = 7x^4 + 2x\)[/tex], we get:
```
3x^3 - 7x^2 + 5 = 7x^4 + 2x
```
This equation directly matches, representing both sides of the system being equal. Therefore, this equation can be solved using the system.
### Option 3: [tex]\( 7x^4 + 2x = 0 \)[/tex]
For this to hold true, we substitute [tex]\(7x^4 + 2x\)[/tex] directly into the second equation. As a result, y = 0 must also hold true. Similar to option 1, it doesn't necessarily represent all solutions of the system.
### Option 4: [tex]\( 7x^4 + 3x^3 - 7x^2 + 2x + 5 = 0 \)[/tex]
In this option, all terms from both equations are combined into a single equation. However, it doesn’t isolate [tex]\(y\)[/tex] on one side and doesn’t directly correspond to an equation derived from the system. Hence, it does not fully represent the system either.
Given this step-by-step analysis, the equation that can be solved by using the system of equations is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
So the correct answer is:
[tex]\[ 3x^3 - 7x^2 + 5 = 7x^4 + 2x \][/tex]
