Solve the system of equations. Enter your answer as an ordered pair.
[tex]
\begin{array}{l}
2 x+4 y=-8 \\
-3 x+4 y=2
\end{array}
[/tex]
Answer (1)
Subtract the second equation from the first to eliminate the y variable: 5 x = − 10 .
Solve for x: x = − 2 .
Substitute the value of x back into the first equation to solve for y: y = − 1 .
Express the solution as an ordered pair: ( − 2 , − 1 ) .
Explanation
Understanding the Problem We are given a system of two linear equations in two variables, x and y. Our goal is to find the values of x and y that satisfy both equations simultaneously. The solution should be expressed as an ordered pair (x, y).
Setting up the Equations We can solve this system of equations using the elimination method. The equations are:
2 x + 4 y = − 8 (1)
− 3 x + 4 y = 2 (2)
Eliminating y To eliminate the y variable, we subtract equation (2) from equation (1):
( 2 x + 4 y ) − ( − 3 x + 4 y ) = − 8 − 2
2 x + 4 y + 3 x − 4 y = − 10
5 x = − 10
Solving for x Now, we solve for x:
x = 5 − 10
x = − 2
Solving for y Substitute the value of x back into equation (1) to solve for y:
2 ( − 2 ) + 4 y = − 8
− 4 + 4 y = − 8
4 y = − 8 + 4
4 y = − 4
y = 4 − 4
y = − 1
Final Answer Therefore, the solution to the system of equations is the ordered pair (-2, -1).
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 instance, if a company wants to know how many units of a product they need to sell to cover their costs, they can set up a system of equations to represent their revenue and expenses. Solving this system will give them the break-even point, which is the number of units they need to sell to make a profit. Understanding how to solve systems of equations is crucial for making informed decisions in many fields.
