$\left\{\begin{array}{l}4 y+x=0 \\ 4 y^2+x^2=5\end{array}\right.
Answer (1)
Solve the first equation for x : x = − 4 y .
Substitute x into the second equation: 4 y 2 + ( − 4 y ) 2 = 5 .
Simplify and solve for y : y = ± 2 1 .
Substitute y back into x = − 4 y to find x : x = − 2 or x = 2 .
The solutions are ( − 2 , 2 1 ) , ( 2 , − 2 1 ) .
Explanation
Understanding the Problem We are given a system of two equations with two variables, x and y. Equation 1: 4 y + x = 0 Equation 2: 4 y 2 + x 2 = 5 We need to find the values of x and y that satisfy both equations.
Solving for x Solve the first equation for x in terms of y: 4 y + x = 0 , so x = − 4 y .
Substitution Substitute the expression for x from step 2 into the second equation: 4 y 2 + ( − 4 y ) 2 = 5 4 y 2 + 16 y 2 = 5 20 y 2 = 5
Solving for y Simplify the equation from step 3 and solve for y: 20 y 2 = 5 y 2 = 20 5 = 4 1 y = ± 4 1 = ± 2 1
Solving for x Substitute the values of y obtained in step 4 back into the equation x = − 4 y to find the corresponding values of x. If y = 2 1 , then x = − 4 ( 2 1 ) = − 2 .
If y = − 2 1 , then x = − 4 ( − 2 1 ) = 2 .
Verification Verify the solutions by substituting the values of x and y into both original equations. For the solution ( − 2 , 2 1 ) :
Equation 1: 4 ( 2 1 ) + ( − 2 ) = 2 − 2 = 0 (Correct) Equation 2: 4 ( 2 1 ) 2 + ( − 2 ) 2 = 4 ( 4 1 ) + 4 = 1 + 4 = 5 (Correct) For the solution ( 2 , − 2 1 ) :
Equation 1: 4 ( − 2 1 ) + ( 2 ) = − 2 + 2 = 0 (Correct) Equation 2: 4 ( − 2 1 ) 2 + ( 2 ) 2 = 4 ( 4 1 ) + 4 = 1 + 4 = 5 (Correct) Thus, the solutions are ( − 2 , 2 1 ) and ( 2 , − 2 1 ) .
Final Answer The solutions to the system of equations are ( − 2 , 2 1 ) and ( 2 , − 2 1 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, modeling supply and demand in economics, and solving for unknown variables in physics problems. For example, if you have two different investment options with different interest rates and initial investments, you can use a system of equations to determine when the two investments will have the same value. This helps in making informed financial decisions.
