Which system is equivalent to $\left\{\begin{array}{c}3 x^2-4 y^2=25 \\ -6 x^2-2 y^2=11\end{array}\right.$ ?
$\left\{\begin{array}{r}3 x^2-4 y^2=25 \\ 12 x^2+4 y^2=22\end{array}\right.$
$\left\{\begin{aligned} 3 x^2-4 y^2 & =25 \\ -12 x^2+4 y^2 & =22\end{aligned}\right.$
$\left\{\begin{array}{r}6 x^2-8 y^2=25 \\ -6 x^2-2 y^2=11\end{array}\right.$
$\left\{\begin{array}{r}6 x^2-8 y^2=50 \\ -6 x^2-2 y^2=11\end{array}\right.$
Answer (1)
Multiply the first equation by 2: 3 x 2 − 4 y 2 = 25 becomes 6 x 2 − 8 y 2 = 50 .
Check each option to see if it matches the transformed system.
Option 4, { 6 x 2 − 8 y 2 = 50 − 6 x 2 − 2 y 2 = 11 , matches the transformed system.
The equivalent system is { 6 x 2 − 8 y 2 = 50 − 6 x 2 − 2 y 2 = 11
Explanation
Understanding the Problem We are given a system of equations and asked to find an equivalent system from the given options. An equivalent system is one that has the same solution set as the original system. We can manipulate the equations in the original system by multiplying them by constants or adding/subtracting multiples of one equation from another to obtain an equivalent system.
Manipulating the Equations The given system is: 3 x 2 − 4 y 2 = 25 − 6 x 2 − 2 y 2 = 11 We will multiply the first equation by 2 to obtain: 2 ( 3 x 2 − 4 y 2 ) = 2 ( 25 ) 6 x 2 − 8 y 2 = 50 Now we check each of the options to see if it is equivalent to the original system.
Checking Option 1 Option 1: { 3 x 2 − 4 y 2 = 25 12 x 2 + 4 y 2 = 22 This is not equivalent since multiplying the second equation in the original system by -2 gives 12 x 2 + 4 y 2 = − 22 , not 22.
Checking Option 2 Option 2: { 3 x 2 − 4 y 2 = 25 − 12 x 2 + 4 y 2 = 22 This is not equivalent since multiplying the second equation in the original system by 2 gives − 12 x 2 − 4 y 2 = 22 , not − 12 x 2 + 4 y 2 = 22 .
Checking Option 3 Option 3: { 6 x 2 − 8 y 2 = 25 − 6 x 2 − 2 y 2 = 11 This is not equivalent since multiplying the first equation in the original system by 2 gives 6 x 2 − 8 y 2 = 50 , not 25.
Checking Option 4 Option 4: { 6 x 2 − 8 y 2 = 50 − 6 x 2 − 2 y 2 = 11 This is equivalent since the first equation is obtained by multiplying the first equation in the original system by 2.
Final Answer Therefore, the equivalent system is: { 6 x 2 − 8 y 2 = 50 − 6 x 2 − 2 y 2 = 11
Examples
Understanding equivalent systems of equations is crucial in various fields, such as physics and engineering, where multiple equations describe the relationships between different variables. For instance, in circuit analysis, you might have a system of equations representing the voltages and currents in different parts of the circuit. Finding an equivalent system can simplify the analysis and make it easier to solve for the unknown variables. This technique is also used in economics to model supply and demand curves and find equilibrium points.
