Use the substitution method to solve the system of equations:

[tex]
\begin{array}{l}
x+y=12 \\
-x=-y-10
\end{array}
[/tex]

A. $(8,4)$
B. $(9,3)$
C. $(10,2)$
D. $(11,1)

Solve the first equation for x : x = 12 − y .
Substitute this expression into the second equation: − ( 12 − y ) = − y − 10 .
Simplify and solve for y : y = 1 .
Substitute y = 1 back into x = 12 − y to find x = 11 . The solution is ( 11 , 1 ) ​ .

Explanation

Problem Analysis We are given the following system of equations:

x + y = 12 − x = − y − 10 ​
Our goal is to solve this system using the substitution method and identify the correct ordered pair from the given options.

Solving for x First, let's solve the first equation for x :
x = 12 − y This expresses x in terms of y .

Substitution Now, substitute this expression for x into the second equation: − ( 12 − y ) = − y − 10 This replaces x in the second equation with the expression we found.

Solving for y Next, simplify and solve for y :
− 12 + y = − y − 10 Add y to both sides: − 12 + 2 y = − 10 Add 12 to both sides: 2 y = 2 Divide by 2 :
y = 1 So, we have found the value of y .

Solving for x Substitute the value of y back into the equation x = 12 − y to find x :
x = 12 − 1 = 11 Thus, we have found the value of x .

Finding the Correct Option The solution is the ordered pair ( x , y ) = ( 11 , 1 ) . Now, we compare this solution with the given options:


A. ( 8 , 4 ) B. ( 9 , 3 ) C. ( 10 , 2 ) D. ( 11 , 1 )
The correct ordered pair is ( 11 , 1 ) , which corresponds to option D.
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For instance, if a company's cost function is C = 5 x + 1000 and its revenue function is R = 15 x , solving the system allows us to find the number of units x needed for the company to break even ( C = R ). This helps in making informed business decisions.

Answered by GinnyAnswer | 2025-07-08

Using the substitution method, we find that the solution to the system of equations is (11, 1). This corresponds to option D. Thus, the answer is D.
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Answered by Anonymous | 2025-07-15