$\begin{array}{r}8 x+y=4 \\ -2 x-8=y\end{array}$
Answer (2)
Substitute the expression for y from the second equation into the first equation: 8 x + ( − 2 x − 8 ) = 4 .
Simplify and solve for x : 6 x = 12 , so x = 2 .
Substitute the value of x back into the second equation to solve for y : y = − 2 ( 2 ) − 8 = − 12 .
The solution to the system of equations is ( 2 , − 12 ) .
Explanation
Understanding the Problem We are given a system of two linear equations:
8 x + y = 4 − 2 x − 8 = y
Our goal is to find the values of x and y that satisfy both equations.
Substitution We can use the substitution method to solve this system. Since the second equation is already solved for y , we can substitute the expression for y from the second equation into the first equation:
8 x + ( − 2 x − 8 ) = 4
Solving for x Now, we simplify and solve for x :
8 x − 2 x − 8 = 4 6 x − 8 = 4 6 x = 4 + 8 6 x = 12 x = 6 12 x = 2
Solving for y Now that we have the value of x , we can substitute it back into either of the original equations to solve for y . Let's use the second equation:
− 2 x − 8 = y − 2 ( 2 ) − 8 = y − 4 − 8 = y y = − 12
Final Answer Therefore, the solution to the system of equations is x = 2 and y = − 12 . We can write this as an ordered pair ( 2 , − 12 ) .
Examples
Systems of equations are used in many real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling traffic flow. For example, a company might use a system of equations to determine how many units of a product they need to sell to cover their costs and start making a profit. Understanding how to solve systems of equations is a valuable skill for anyone interested in business, science, or engineering.
The solution to the system of equations is x = 2 and y = − 12 , given as the ordered pair ( 2 , − 12 ) . This solution satisfies both equations simultaneously.
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