Solve this system of equations
$\left\{\begin{array}{lll}
2 x-4 y & = & -16 \\
-4 x-4 y & = & -16
\end{array}\right.$
Answer (1)
Subtract the second equation from the first to eliminate y : 6 x = 0 .
Solve for x : x = 0 .
Substitute x = 0 into the first equation: − 4 y = − 16 .
Solve for y : y = 4 . The solution is x = 0 , y = 4 .
Explanation
Problem Analysis We are given 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 simultaneously. The given equations are:
Given Equations 2 x − 4 y = − 16 − 4 x − 4 y = − 16
Solution Strategy We can solve this system using the elimination method. Notice that the coefficients of y in both equations are the same (-4). We can subtract the second equation from the first to eliminate y .
Eliminating y Subtract the second equation from the first: ( 2 x − 4 y ) − ( − 4 x − 4 y ) = − 16 − ( − 16 ) 2 x − 4 y + 4 x + 4 y = − 16 + 16 6 x = 0
Solving for x Divide both sides by 6 to solve for x : x = 6 0 = 0
Substituting x into the first equation 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: 2 x − 4 y = − 16 2 ( 0 ) − 4 y = − 16 − 4 y = − 16
Solving for y Divide both sides by -4 to solve for y : y = − 4 − 16 = 4
Final Answer Therefore, the solution to the system of equations is x = 0 and y = 4 .
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 products to maximize profit, or modeling supply and demand in economics. For example, suppose a company sells two products, A and B. The cost and revenue equations can form a system of equations, and solving it helps determine the quantity of each product that needs to be sold to break even or achieve a specific profit target. Understanding how to solve systems of equations is crucial for making informed decisions in business and economics.
