Solve the system for $x$ and $y$.
$\begin{array}{l}
6 x-y=\pi \\
8 x-5 y=-71
\end{array}$
A. (2.19)
B. 12.111
C. (9.27)
D. (2.11)
Answer (2)
The system of equations was solved using the elimination method, yielding approximate values of x ≈ 3.935 and y ≈ 20.468 . None of the provided multiple-choice options correspond to this solution. Therefore, the answer does not match any of the given options.
;
Multiply the first equation by 5 to prepare for eliminating y : 30 x − 5 y = 5 π .
Subtract the second equation from the modified first equation: 22 x = 5 π + 71 .
Solve for x : x = 22 5 π + 71 ≈ 3.94 .
Substitute x back into the first equation to solve for y : y = 6 x − π ≈ 20.51 .
No match in given options.
Explanation
Analyze the problem and choose a solving method We are given a system of two linear equations with two variables, x and y :
{ 6 x − y = π 8 x − 5 y = − 71
Our goal is to find the values of x and y that satisfy both equations. We can use either substitution or elimination method to solve this system. Here, we will use the elimination method.
Multiply the first equation by 5 To eliminate y , we multiply the first equation by 5:
5 ( 6 x − y ) = 5 ( π ) 30 x − 5 y = 5 π
Now we have the following system:
{ 30 x − 5 y = 5 π 8 x − 5 y = − 71
Subtract the equations and solve for x Subtract the second equation from the modified first equation to eliminate y :
( 30 x − 5 y ) − ( 8 x − 5 y ) = 5 π − ( − 71 ) 30 x − 5 y − 8 x + 5 y = 5 π + 71 22 x = 5 π + 71
Now, solve for x :
x = 22 5 π + 71 x ≈ 22 5 ( 3.14159 ) + 71 ≈ 22 15.708 + 71 ≈ 22 86.708 ≈ 3.941
Substitute x into the first equation and solve for y Substitute the value of x back into the first original equation to solve for y :
6 x − y = π 6 ( 3.941 ) − y = π 23.646 − y = 3.14159 y = 23.646 − 3.14159 y ≈ 20.504
State the solution Therefore, the solution to the system of equations is approximately:
x ≈ 3.94 y ≈ 20.51
None of the options match the solution.
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling supply and demand in economics. For example, a company might use a system of equations to determine the number of units they need to sell to cover their costs and start making a profit. By setting up equations that represent their costs and revenue, they can solve for the quantity at which these two are equal, giving them the break-even point.
