Solve the following system of equations by the substitution method. Check the solutions.
[tex]\left\{\begin{aligned}
25 x^2+9 y^2 & =225 \\
5 x+3 y & =15
\end{aligned}\right.[/tex]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is { }. (Type an ordered pair. Use a comma to separate answers as needed.)
B. There is no solution.
Answer (1)
Solve the second equation for y : y = 5 − 3 5 x .
Substitute this expression into the first equation and simplify: 50 x 2 − 150 x = 0 .
Factor and solve for x : x = 0 or x = 3 .
Substitute the values of x back into the equation for y to find the corresponding y values. The solution set is $\boxed{{(0, 5), (3, 0)}}.
Explanation
Analyze the problem We are given the following system of equations:
{ 25 x 2 + 9 y 2 = 225 5 x + 3 y = 15
We need to solve this system using the substitution method and check our solutions.
Solve for y First, let's solve the second equation for y in terms of x :
5 x + 3 y = 15 3 y = 15 − 5 x y = 3 15 − 5 x y = 5 − 3 5 x
Substitute into the first equation Now, substitute this expression for y into the first equation:
25 x 2 + 9 ( 5 − 3 5 x ) 2 = 225
Expand and simplify Expand and simplify the equation:
25 x 2 + 9 ( 25 − 3 50 x + 9 25 x 2 ) = 225 25 x 2 + 225 − 150 x + 25 x 2 = 225 50 x 2 − 150 x = 0
Factor Factor the equation:
50 x ( x − 3 ) = 0
Solve for x Solve for x :
50 x = 0 or x − 3 = 0 x = 0 or x = 3
Solve for y For each value of x , find the corresponding value of y using the equation y = 5 − 3 5 x .
If x = 0 , then
y = 5 − 3 5 ( 0 ) = 5
If x = 3 , then
y = 5 − 3 5 ( 3 ) = 5 − 5 = 0
Check the solutions The solutions are ( 0 , 5 ) and ( 3 , 0 ) .
Check the solutions in both original equations.
For ( 0 , 5 ) :
25 ( 0 ) 2 + 9 ( 5 ) 2 = 0 + 225 = 225 5 ( 0 ) + 3 ( 5 ) = 0 + 15 = 15
For ( 3 , 0 ) :
25 ( 3 ) 2 + 9 ( 0 ) 2 = 25 ( 9 ) + 0 = 225 5 ( 3 ) + 3 ( 0 ) = 15 + 0 = 15
Both solutions satisfy both equations.
Final Answer The solution set is {(0, 5), (3, 0)}.
Examples
Systems of equations are used in various real-world applications, such as determining the optimal mix of products to maximize profit, calculating the trajectory of a projectile, or modeling supply and demand in economics. In this case, the given system could represent the constraints on the production of two goods, where x and y represent the quantities of each good produced. Solving the system helps determine the feasible production levels that satisfy both constraints.
