What are the solutions to the following system?
$\begin{array}{l}
-2 x^2+y=-5 \\
y=-3 x^2+5
\end{array}$
A. $(\sqrt{5},-10)$ and $(-\sqrt{5},-10)$
B. $(0,2)$
C. $(1,-2)$
D. $(\sqrt{2},-1)$ and $(-\sqrt{2},-1)$
Answer (1)
Substitute the second equation into the first equation.
Simplify and solve for x 2 , obtaining x = ± 2 .
Substitute x = 2 and x = − 2 into the second equation to find the corresponding y values.
The solutions are ( 2 , − 1 ) and ( − 2 , − 1 ) .
( 2 , − 1 ) and ( − 2 , − 1 )
Explanation
Analyze the problem We are given a system of two equations with two variables, x and y :
− 2 x 2 + y = − 5 y = − 3 x 2 + 5
Our goal is to find the solutions ( x , y ) that satisfy both equations. We can use the substitution method to solve this system.
Substitution Substitute the second equation, y = − 3 x 2 + 5 , into the first equation: − 2 x 2 + y = − 5 − 2 x 2 + ( − 3 x 2 + 5 ) = − 5
Simplify Simplify the equation: − 2 x 2 − 3 x 2 + 5 = − 5 − 5 x 2 + 5 = − 5
Isolate x 2 Solve for x 2 :
− 5 x 2 = − 5 − 5 − 5 x 2 = − 10 x 2 = − 5 − 10 x 2 = 2
Solve for x Solve for x :
x = ± 2
Find y for x = 2 Substitute x = 2 into y = − 3 x 2 + 5 to find the corresponding y value: y = − 3 ( 2 ) 2 + 5 y = − 3 ( 2 ) + 5 y = − 6 + 5 y = − 1
Find y for x = − 2 Substitute x = − 2 into y = − 3 x 2 + 5 to find the corresponding y value: y = − 3 ( − 2 ) 2 + 5 y = − 3 ( 2 ) + 5 y = − 6 + 5 y = − 1
Solutions The solutions are ( 2 , − 1 ) and ( − 2 , − 1 ) .
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 electrical engineering, systems of equations can be used to analyze circuits with multiple loops and nodes, determining the currents and voltages in different parts of the circuit. In economics, they can model supply and demand curves to find equilibrium prices and quantities in a market.
