Use the elimination method to find all solutions of the system. Write the solution(s) as a list of ordered pairs.
[tex]\left\{\begin{array}{l}
x^2-2 y=10 \
x^2+3 y=25\end{array}\right.[/tex]
Answer (2)
Eliminate x 2 by subtracting the first equation from the second, resulting in 5 y = 15 .
Solve for y , obtaining y = 3 .
Substitute y = 3 into the first equation to find x 2 = 16 .
Solve for x , yielding x = ± 4 . The solutions are ( 4 , 3 ) , ( − 4 , 3 ) .
Explanation
System of Equations We are given the system of equations: x 2 − 2 y = 10 x 2 + 3 y = 25 We will use the elimination method to solve for x and y .
Eliminate x^2 Subtract the first equation from the second equation to eliminate x 2 :
( x 2 + 3 y ) − ( x 2 − 2 y ) = 25 − 10 x 2 + 3 y − x 2 + 2 y = 15 5 y = 15
Solve for y Solve for y :
y = 5 15 y = 3
Substitute y into equation 1 Substitute y = 3 into the first equation to solve for x :
x 2 − 2 ( 3 ) = 10 x 2 − 6 = 10 x 2 = 16
Solve for x Solve for x :
x = ± 16 x = ± 4
Check the solutions The solutions are ( 4 , 3 ) and ( − 4 , 3 ) .
Check the solutions in both original equations: For ( 4 , 3 ) :
4 2 − 2 ( 3 ) = 16 − 6 = 10 4 2 + 3 ( 3 ) = 16 + 9 = 25 For ( − 4 , 3 ) :
( − 4 ) 2 − 2 ( 3 ) = 16 − 6 = 10 ( − 4 ) 2 + 3 ( 3 ) = 16 + 9 = 25 Both solutions satisfy both equations.
Final Answer The solutions to the system of equations are ( 4 , 3 ) and ( − 4 , 3 ) .
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, or calculating the trajectory of a projectile in physics. In this case, we found the points where two curves (defined by the given equations) intersect. This concept can be applied to optimize resource allocation or predict outcomes in complex scenarios.
Using the elimination method, we find the solutions to the system of equations are the ordered pairs (4, 3) and (-4, 3). These solutions are verified by substituting them back into the original equations. Thus, the final answer comprises the points where the two equations intersect.
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