\begin{array}{l}-5 x+1 y=-3 \\ 3 x-8 y=24\end{array}

Solve the first equation for y : y = 5 x − 3 .
Substitute this expression for y into the second equation and solve for x : x = 0 .
Substitute the value of x back into the equation for y and solve for y : y = − 3 .
The solution to the system of equations is x = 0 , y = − 3 ​ .

Explanation

Analyze the problem We are given a system of two linear equations:

− 5 x + y = − 3 3 x − 8 y = 24
Our goal is to find the values of x and y that satisfy both equations. We will use the substitution method to solve this system.

Solve for y First, we solve the first equation for y in terms of x :

y = 5 x − 3

Substitute y in second equation Next, we substitute this expression for y into the second equation:

3 x − 8 ( 5 x − 3 ) = 24

Solve for x Now, we simplify and solve the resulting equation for x :

3 x − 40 x + 24 = 24 − 37 x = 0 x = 0

Solve for y Now that we have the value of x , we substitute it back into the equation y = 5 x − 3 to find the value of y :

y = 5 ( 0 ) − 3 y = − 3

State the solution Therefore, the solution to the system of equations is x = 0 and y = − 3 .

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 company produces two products, A and B. Each product requires different amounts of labor and materials. By setting up a system of equations, the company can determine the optimal production levels for each product to maximize profit, given constraints on available labor and materials. Understanding how to solve systems of equations is crucial for making informed decisions in many fields.

Answered by GinnyAnswer | 2025-07-08