Which system is equivalent to $\left\{\begin{array}{l}5 x^2+6 y^2=50 \\ 7 x^2+2 y^2=10\end{array} ?\right. $\left\{\begin{array}{r}5 x^2+6 y^2=50 \\ -21 x^2-6 y^2=10\end{array}\right. $\left\{\begin{aligned} 5 x^2+0 y^2 & =50 \\ -21 x^2-6 y^2 & =30\end{aligned}\right. $\left\{\begin{aligned} 35 x^2+42 y^2 & =250 \\ -35 x^2-10 y^2 & =-50\end{aligned}\right. p. $\left\{\begin{array}{l}35 x^2+12 y^2=350 \\ -35 x^2-10 y^2=-50\end{array}\right.
Answer (2)
The equivalent system to the given equations is { 35 x 2 + 42 y 2 = 350 − 35 x 2 − 10 y 2 = − 50 . This was derived by manipulating the original equations through multiplication. The choice corresponds to the correct option presented in the question.
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Multiply the second equation by -3: − 21 x 2 − 6 y 2 = − 30 .
The equivalent system is { 5 x 2 + 6 y 2 = 50 − 21 x 2 − 6 y 2 = − 30 .
Multiply the first equation by 7 and the second equation by -5 to obtain another equivalent system: { 35 x 2 + 42 y 2 = 350 − 35 x 2 − 10 y 2 = − 50 .
The equivalent system is { 35 x 2 + 42 y 2 = 350 − 35 x 2 − 10 y 2 = − 50 .
Explanation
Analyzing the System of Equations We are given a system of two equations:
{ 5 x 2 + 6 y 2 = 50 7 x 2 + 2 y 2 = 10
Our goal is to find an equivalent system among the given options. An equivalent system is one that has the same solutions as the original system. We can obtain an equivalent system by multiplying one or both equations by a constant, or by adding a multiple of one equation to another.
Creating an Equivalent System Let's multiply the second equation by -3: − 3 ( 7 x 2 + 2 y 2 ) = − 3 ( 10 ) − 21 x 2 − 6 y 2 = − 30 Now we have the system: { 5 x 2 + 6 y 2 = 50 − 21 x 2 − 6 y 2 = − 30 This system is equivalent to the original system.
Creating Another Equivalent System Let's multiply the first equation of the original system by 7: 7 ( 5 x 2 + 6 y 2 ) = 7 ( 50 ) 35 x 2 + 42 y 2 = 350 Now multiply the second equation of the original system by -5: − 5 ( 7 x 2 + 2 y 2 ) = − 5 ( 10 ) − 35 x 2 − 10 y 2 = − 50 So we have another equivalent system: { 35 x 2 + 42 y 2 = 350 − 35 x 2 − 10 y 2 = − 50
Comparing with the Options Now, let's compare the equivalent systems we found with the given options:
Option 1: { 5 x 2 + 6 y 2 = 50 − 21 x 2 − 6 y 2 = 10 The second equation is incorrect.
Option 2: { 5 x 2 + 0 y 2 − 21 x 2 − 6 y 2 = 50 = 30 Both equations are incorrect.
Option 3: { 35 x 2 + 42 y 2 − 35 x 2 − 10 y 2 = 250 = − 50 The first equation is incorrect.
Option 4: { 35 x 2 + 42 y 2 = 350 − 35 x 2 − 10 y 2 = − 50 This system matches the one we derived.
Final Answer Therefore, the equivalent system is: { 35 x 2 + 42 y 2 = 350 − 35 x 2 − 10 y 2 = − 50
Examples
Systems of equations are used in various fields, such as physics, engineering, and economics, to model and solve problems involving multiple variables and constraints. For example, in circuit analysis, systems of equations can be used to determine the currents and voltages in different parts of a circuit. In economics, they can be used to model the supply and demand of goods and services. Understanding how to manipulate and solve systems of equations is crucial for tackling these real-world problems.
