Solve the system of equations given below.

[tex]
\begin{array}{l}
-5 x=y-5 \\
-2 y=-x-21
\end{array}
[/tex]

A. (10,16)
B. (-1,-23)
C. (10,-45)
D. (-1,10)

Rewrite the equations in standard form: 5 x + y = 5 and x − 2 y = 21 .
Solve for x and y using substitution or elimination. The correct solution is x = 11 31 ​ and y = 11 − 100 ​ .
Verify the given options. None of the options match the correct solution.
Conclude that there might be an error in the provided options or the original equations. Based on the original equations, none of the answers are correct.

Explanation

Problem Analysis We are given a system of two linear equations and asked to find the solution (x, y) that satisfies both equations. The equations are:

Equation 1: − 5 x = y − 5 Equation 2: − 2 y = − x − 21

Rewriting Equations First, let's rewrite the equations in the standard form A x + B y = C :

Equation 1: − 5 x = y − 5 ⇒ 5 x + y = 5 Equation 2: − 2 y = − x − 21 ⇒ x − 2 y = 21

Expressing y in terms of x Now, we can solve the system of equations using either substitution or elimination. Let's use the substitution method. From Equation 1, we can express y in terms of x :

y = 5 − 5 x

Solving for x Substitute this expression for y into Equation 2:

x − 2 ( 5 − 5 x ) = 21 x − 10 + 10 x = 21 11 x = 31 x = 11 31 ​

Solving for y Now, substitute the value of x back into the expression for y :

y = 5 − 5 ( 11 31 ​ ) y = 5 − 11 155 ​ y = 11 55 − 155 ​ y = 11 − 100 ​

Checking the Options and Re-evaluating So, the solution to the system of equations is x = 11 31 ​ and y = 11 − 100 ​ . Now we need to check which of the given options matches our solution. Since our calculated solution doesn't match any of the provided options, let's re-examine the elimination method to ensure accuracy.

Using Elimination Method Let's use the elimination method. We have:


Equation 1: 5 x + y = 5 Equation 2: x − 2 y = 21
Multiply Equation 1 by 2 to eliminate y :
10 x + 2 y = 10
Add this to Equation 2:
( 10 x + 2 y ) + ( x − 2 y ) = 10 + 21 11 x = 31 x = 11 31 ​

Solving for y Again Substitute x = 11 31 ​ into Equation 1:

5 ( 11 31 ​ ) + y = 5 11 155 ​ + y = 5 y = 5 − 11 155 ​ y = 11 55 − 155 ​ y = 11 − 100 ​

Verifying the Options The solution we found ( x = 11 31 ​ , y = 11 − 100 ​ ) still doesn't match any of the given options (A. ( 10 , 16 ) , B. ( − 1 , − 23 ) , C. ( 10 , − 45 ) , D. ( − 1 , 10 ) ). It seems there might be an error in the provided options or in my calculations. Let's verify the options by substituting them into the original equations.

Option A: ( 10 , 16 ) Equation 1: 5 ( 10 ) + 16 = 50 + 16 = 66  = 5
Option B: ( − 1 , − 23 ) Equation 1: 5 ( − 1 ) + ( − 23 ) = − 5 − 23 = − 28  = 5
Option C: ( 10 , − 45 ) Equation 1: 5 ( 10 ) + ( − 45 ) = 50 − 45 = 5 (Correct for Equation 1) Equation 2: 10 − 2 ( − 45 ) = 10 + 90 = 100  = 21
Option D: ( − 1 , 10 ) Equation 1: 5 ( − 1 ) + 10 = − 5 + 10 = 5 (Correct for Equation 1) Equation 2: − 1 − 2 ( 10 ) = − 1 − 20 = − 21  = 21 . However, if the equation was x − 2 y = − 21 , then − 1 − 2 ( 10 ) = − 21 would be correct.
Given the original equations, none of the options are correct. However, if the second equation was − 2 y = − x − 21 ⇒ x − 2 y = − 21 , then option D would be correct.

Final Conclusion Since none of the provided options satisfy the given system of equations, and after double-checking the calculations, it appears there might be a typo in the problem or the options. Assuming the second equation is actually − 2 y = − x − 21 ⇒ x − 2 y = − 21 , then option D ( − 1 , 10 ) would be the correct answer. However, based on the original problem, none of the answers are correct.

Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, a company might want to know how many units they need to sell to cover their costs. By setting up equations for cost and revenue, they can solve for the number of units needed to break even. This helps in making informed business decisions.

Answered by GinnyAnswer | 2025-07-07

The solution to the system of equations is (-1, 10), which corresponds to Option D. Both equations are satisfied by this solution, confirming its accuracy. Thus, D is the correct answer.
;

Answered by Anonymous | 2025-08-16