$6 x+10 y=17000$
$3 x+22 y=17000$
Find $x$ and $y$ using the elimination method in a simultaneous equation.

By using the elimination method on the given system of equations, we find that x = 2000 and y = 500 . We first equalized the coefficients, eliminated one variable, and solved for the other. Finally, we substituted back to get the complete solution.
;

Answered by Anonymous | 2025-07-04

Multiply the second equation by 2 to make the coefficients of x equal: 6 x + 44 y = 34000 .
Subtract the first equation from the modified second equation to eliminate x : ( 6 x + 44 y ) − ( 6 x + 10 y ) = 34000 − 17000 .
Simplify and solve for y : 34 y = 17000 , so y = 500 .
Substitute the value of y into the first equation and solve for x : 6 x + 10 ( 500 ) = 17000 , so x = 2000 .

The solution is x = 2000 , y = 500 ​ .
Explanation

Problem Analysis We are given a system of two equations with two variables, x and y . Our goal is to find the values of x and y that satisfy both equations simultaneously using the elimination method. The equations are:


6 x + 10 y = 17000
3 x + 22 y = 17000


Preparing for Elimination To use the elimination method, we want to make the coefficients of either x or y the same in both equations so that we can eliminate one variable by subtracting one equation from the other. In this case, it's easier to make the coefficients of x the same. We can multiply the second equation by 2:

2 × ( 3 x + 22 y ) = 2 × 17000 6 x + 44 y = 34000

Eliminating x Now we have the following system of equations:


6 x + 10 y = 17000
6 x + 44 y = 34000

We can subtract the first equation from the second equation to eliminate x :
( 6 x + 44 y ) − ( 6 x + 10 y ) = 34000 − 17000

Simplifying the Equation Simplifying the equation, we get:

6 x + 44 y − 6 x − 10 y = 17000 34 y = 17000

Solving for y Now, we can solve for y :

y = 34 17000 ​ y = 500

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

6 x + 10 y = 17000 6 x + 10 ( 500 ) = 17000 6 x + 5000 = 17000

Solving for x Now, we can solve for x :

6 x = 17000 − 5000 6 x = 12000 x = 6 12000 ​ x = 2000

Final Answer Therefore, the solution to the system of equations is x = 2000 and y = 500 .

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 use a system of equations to model its costs and revenues, and then solve the system to find the level of production at which its costs equal its revenues. This helps the company make informed decisions about pricing, production volume, and other factors that affect its profitability. Another example is in electrical engineering, where systems of equations are used to analyze circuits and determine the currents and voltages at different points in the circuit.

Answered by GinnyAnswer | 2025-07-04