Solve the system by the addition method.
[tex]\begin{array}{l}
x+y=6 \
x-y=4\end{array}[/tex]

Select the correct choice below and fill in any answer boxes present in your choice.
A. The solution set is $\square$ \}. (Simplify your answer. Type an ordered pair.)
B. There are infinitely many solutions.
C. There is no solution.

Add the two equations to eliminate y : 2 x = 10 .
Solve for x : x = 5 .
Substitute the value of x into one of the original equations to solve for y : 5 + y = 6 , so y = 1 .
The solution set is { ( 5 , 1 ) } ​ .

Explanation

Analyze the problem We are given a system of two linear equations with two variables, x and y :

x + y = 6 x − y = 4 ​
Our goal is to solve this system using the addition method, which involves adding the equations in such a way that one of the variables is eliminated.

Eliminate y and solve for x To eliminate y , we can simply add the two equations:

( x + y ) + ( x − y ) = 6 + 4
This simplifies to:
2 x = 10
Now, we can solve for x by dividing both sides by 2:
x = 2 10 ​ = 5
So, x = 5 .

Solve for y Now that we have the value of x , we can substitute it into either of the original equations to solve for y . Let's use the first equation:

x + y = 6
Substitute x = 5 :
5 + y = 6
Subtract 5 from both sides to solve for y :
y = 6 − 5 = 1
So, y = 1 .

State the solution The solution to the system of equations is the ordered pair ( x , y ) = ( 5 , 1 ) .

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, suppose a bakery sells cookies and cakes. Let x be the number of cookies and y be the number of cakes. If the total revenue is given by x + y = 6 (in thousands of dollars) and the difference in production cost is x − y = 4 (in thousands of dollars), solving this system helps the bakery determine the optimal production levels for cookies and cakes to meet their financial goals.

Answered by GinnyAnswer | 2025-07-03

The solution to the system of equations is the ordered pair (5, 1). Therefore, the solution set is {( 5 , 1 )} .
;

Answered by Anonymous | 2025-07-04