Answer :
To solve the problem of finding an equivalent system for the given equations, we start with the two original equations:
1) [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2) [tex]\(7x^2 + 2y^2 = 10\)[/tex]
Our goal is to find a set of operations that transforms these equations into one of the given options.
Step 1: Eliminate terms using multiplication
- Multiply the entire first equation by 7:
[tex]\[
7 \times (5x^2 + 6y^2) = 7 \times 50
\][/tex]
This gives:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]
- Multiply the entire second equation by -5:
[tex]\[
-5 \times (7x^2 + 2y^2) = -5 \times 10
\][/tex]
This gives:
[tex]\[
-35x^2 - 10y^2 = -50
\][/tex]
Step 2: Write down the new system of equations
After the operations, the new system of equations is:
- First equation: [tex]\(35x^2 + 42y^2 = 350\)[/tex]
- Second equation: [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
This matches the fourth option given in the problem:
[tex]\[
\left\{\begin{aligned}
35x^2 + 42y^2 & = 350 \\
-35x^2 - 10y^2 & = -50
\end{aligned}\right.
\][/tex]
Thus, the equivalent system is:
- [tex]\(35x^2 + 42y^2 = 350\)[/tex]
- [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
This methodically illustrates how the solution to the equivalent system can be derived using multiplication and subtraction to eliminate terms, leading to the correct equivalent system.
1) [tex]\(5x^2 + 6y^2 = 50\)[/tex]
2) [tex]\(7x^2 + 2y^2 = 10\)[/tex]
Our goal is to find a set of operations that transforms these equations into one of the given options.
Step 1: Eliminate terms using multiplication
- Multiply the entire first equation by 7:
[tex]\[
7 \times (5x^2 + 6y^2) = 7 \times 50
\][/tex]
This gives:
[tex]\[
35x^2 + 42y^2 = 350
\][/tex]
- Multiply the entire second equation by -5:
[tex]\[
-5 \times (7x^2 + 2y^2) = -5 \times 10
\][/tex]
This gives:
[tex]\[
-35x^2 - 10y^2 = -50
\][/tex]
Step 2: Write down the new system of equations
After the operations, the new system of equations is:
- First equation: [tex]\(35x^2 + 42y^2 = 350\)[/tex]
- Second equation: [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
This matches the fourth option given in the problem:
[tex]\[
\left\{\begin{aligned}
35x^2 + 42y^2 & = 350 \\
-35x^2 - 10y^2 & = -50
\end{aligned}\right.
\][/tex]
Thus, the equivalent system is:
- [tex]\(35x^2 + 42y^2 = 350\)[/tex]
- [tex]\(-35x^2 - 10y^2 = -50\)[/tex]
This methodically illustrates how the solution to the equivalent system can be derived using multiplication and subtraction to eliminate terms, leading to the correct equivalent system.
