$(5x-3y, 3x-y) = (16, 12)

Set up the system of equations: 5 x − 3 y = 16 and 3 x − y = 12 .
Solve the second equation for y : y = 3 x − 12 .
Substitute the expression for y into the first equation and solve for x : x = 5 .
Substitute the value of x back into the equation for y and solve: y = 3 . The solution is x = 5 , y = 3 ​ .

Explanation

Understanding the Problem We are given the equation ( 5 x − 3 y , 3 x − y ) = ( 16 , 12 ) . This represents a system of two linear equations with two variables, x and y . Our goal is to find the values of x and y that satisfy both equations.

Setting up the Equations We can set up the following system of equations:


5 x − 3 y = 16 3 x − y = 12

Solving for y in the Second Equation We can solve this system of equations using either substitution or elimination. Let's use the substitution method. From the second equation, we can express y in terms of x :

y = 3 x − 12

Substituting into the First Equation Now, substitute this expression for y into the first equation:

5 x − 3 ( 3 x − 12 ) = 16

Solving for x Simplify and solve for x :

5 x − 9 x + 36 = 16 − 4 x = 16 − 36 − 4 x = − 20 x = − 4 − 20 ​ x = 5

Solving for y Now that we have the value of x , we can substitute it back into the equation y = 3 x − 12 to find the value of y :

y = 3 ( 5 ) − 12 y = 15 − 12 y = 3

Final Answer Therefore, the solution to the system of equations is x = 5 and y = 3 .

Examples
Systems of equations are incredibly useful in real life. For example, imagine you're running a small business that sells two products. You know the total revenue you made last month and the total number of products you sold. By setting up a system of equations, you can determine the price of each product. This helps in pricing strategies, inventory management, and understanding your sales data better.

Answered by GinnyAnswer | 2025-07-08