Answer :
To solve this problem, we need to find an equivalent system of equations to the given one:
Original system:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(7x^2 + 2y^2 = 10\)[/tex]
To find an equivalent system, we can scale and manipulate these equations. We'll proceed as follows:
1. Scale the equations to eliminate a variable:
- Multiply the first equation by 7:
[tex]\[
7(5x^2 + 6y^2) = 7 \times 50
\][/tex]
This gives:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]
- Multiply the second equation by 5:
[tex]\[
5(7x^2 + 2y^2) = 5 \times 10
\][/tex]
This gives:
[tex]\[
35x^2 + 10y^2 = 50
\][/tex]
2. Subtract the second scaled equation from the first scaled equation:
- Subtraction:
[tex]\[
(35x^2 + 42y^2) - (35x^2 + 10y^2) = 350 - 50
\][/tex]
Simplify the left side:
[tex]\[
42y^2 - 10y^2 = 32y^2
\][/tex]
The right side:
[tex]\[
350 - 50 = 300
\][/tex]
So the equation simplifies to:
[tex]\[
32y^2 = 300
\][/tex]
3. Check and simplify the system:
- We derived the system:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]
[tex]\[
-35x^2 - 10y^2 = -50
\][/tex]
After these steps, the equations form the equivalent system:
1. [tex]\(35x^2 + 42y^2 = 350\)[/tex]
2. [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
Thus, the equivalent system corresponds to the fourth option:
[tex]\[
\left\{\begin{aligned}
35x^2+42y^2 & = 350 \\
-35x^2-10y^2 & = -50
\end{aligned}\right.
\][/tex]
Original system:
1. [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2. [tex]\(7x^2 + 2y^2 = 10\)[/tex]
To find an equivalent system, we can scale and manipulate these equations. We'll proceed as follows:
1. Scale the equations to eliminate a variable:
- Multiply the first equation by 7:
[tex]\[
7(5x^2 + 6y^2) = 7 \times 50
\][/tex]
This gives:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]
- Multiply the second equation by 5:
[tex]\[
5(7x^2 + 2y^2) = 5 \times 10
\][/tex]
This gives:
[tex]\[
35x^2 + 10y^2 = 50
\][/tex]
2. Subtract the second scaled equation from the first scaled equation:
- Subtraction:
[tex]\[
(35x^2 + 42y^2) - (35x^2 + 10y^2) = 350 - 50
\][/tex]
Simplify the left side:
[tex]\[
42y^2 - 10y^2 = 32y^2
\][/tex]
The right side:
[tex]\[
350 - 50 = 300
\][/tex]
So the equation simplifies to:
[tex]\[
32y^2 = 300
\][/tex]
3. Check and simplify the system:
- We derived the system:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]
[tex]\[
-35x^2 - 10y^2 = -50
\][/tex]
After these steps, the equations form the equivalent system:
1. [tex]\(35x^2 + 42y^2 = 350\)[/tex]
2. [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
Thus, the equivalent system corresponds to the fourth option:
[tex]\[
\left\{\begin{aligned}
35x^2+42y^2 & = 350 \\
-35x^2-10y^2 & = -50
\end{aligned}\right.
\][/tex]
