Solve the system for x and y.
[tex]
\begin{array}{l}
-8 x-y=-43 \\
8 x-5 y=-71
\end{array}
[/tex]
A. (3,19)
B. (-2,11)
C. (2,27)
D. (4,11)
Answer (2)
The solution to the given system of equations is (3, 19). This was found by using the elimination method to solve for y first, then substituting back to find x. Thus, the answer choice is (3, 19).
;
Add the two equations to eliminate x : ( − 8 x − y ) + ( 8 x − 5 y ) = − 43 + ( − 71 ) .
Simplify and solve for y : − 6 y = − 114 , so y = 19 .
Substitute y = 19 into the first equation: − 8 x − 19 = − 43 , which gives − 8 x = − 24 .
Solve for x : x = 3 . The solution is ( 3 , 19 ) .
Explanation
Analyze the problem We are given a system of two linear equations with two variables, x and y:
Equation 1: − 8 x − y = − 43 Equation 2: 8 x − 5 y = − 71
Our objective is to find the values of x and y that satisfy both equations. We can solve this system using the method of elimination.
Eliminate x and solve for y Add the two equations to eliminate x:
( − 8 x − y ) + ( 8 x − 5 y ) = − 43 + ( − 71 )
Simplify the resulting equation:
− 6 y = − 114
Solve for y:
y = − 6 − 114 = 19
Solve for x Substitute the value of y back into either Equation 1 or Equation 2 to solve for x. Let's use Equation 1:
− 8 x − 19 = − 43
Add 19 to both sides:
− 8 x = − 43 + 19
− 8 x = − 24
Divide by -8:
x = − 8 − 24 = 3
State the final answer Therefore, the solution to the system of equations is x = 3 and y = 19 . So the answer is ( 3 , 19 ) .
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 understanding how to solve systems of equations, students can apply these concepts to solve practical problems in various fields.
