$2x+y=8$ and $x-y=1$

To solve the system of equations 2 x + y = 8 and x − y = 1 , we used the elimination method to find x = 3 and y = 2 . This solution satisfies both equations, which we verified. The final answer is x = 3 , y = 2 ​ .
;

Answered by Anonymous | 2025-07-04

Add the two equations to eliminate y : 3 x = 9 .
Solve for x : x = 3 .
Substitute x = 3 into x − y = 1 to solve for y : 3 − y = 1 , so y = 2 .
The solution is x = 3 , y = 2 ​ .

Explanation

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


2 x + y = 8
x − y = 1

Our goal is to find the values of x and y that satisfy both equations simultaneously. We can use either the substitution or elimination method to solve this system.

Eliminate y and solve for x Let's use the elimination method. We can add the two equations together to eliminate the variable y :

( 2 x + y ) + ( x − y ) = 8 + 1
Combining like terms, we get:
3 x = 9
Now, divide both sides by 3 to solve for x :
x = 3 9 ​ = 3

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

x − y = 1
Substitute x = 3 :
3 − y = 1
Subtract 3 from both sides:
− y = 1 − 3
− y = − 2
Multiply both sides by -1:
y = 2

Verify the solution We have found that x = 3 and y = 2 . Let's check our solution by substituting these values into both original equations:


2 x + y = 2 ( 3 ) + 2 = 6 + 2 = 8 (Correct)
x − y = 3 − 2 = 1 (Correct)

Since our solution satisfies both equations, it is the correct solution.

State the final answer Therefore, the solution to the system of equations is x = 3 and y = 2 .

x = 3 , y = 2 ​
Examples
Systems of equations are used in various real-life scenarios. For example, when planning a party, you might need to determine the number of pizzas and drinks to buy based on the number of guests and their preferences, given a budget constraint. Similarly, in business, systems of equations can help determine the optimal production levels of different products to maximize profit, considering resource limitations and market demand. In physics, they can be used to analyze the forces acting on an object in equilibrium.

Answered by GinnyAnswer | 2025-07-04