$2 x+y=8$ and $x-y=1$. Solve by substitution method.
Answer (2)
To solve the equations 2 x + y = 8 and x − y = 1 using substitution, first solve for x in terms of y , substitute into the other equation, and solve for y . This gives y = 2 , which allows you to find x = 3 . The solution is the ordered pair ( 3 , 2 ) .
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Solve the second equation for x : x = y + 1 .
Substitute x into the first equation: 2 ( y + 1 ) + y = 8 .
Solve for y : y = 2 .
Substitute y back to find x : x = 3 . The solution is ( 3 , 2 ) .
Explanation
Analyze the problem We are given the following system of equations:
2 x + y = 8 x − y = 1
We need to solve this system using the substitution method.
Solve for x First, we solve the second equation for x in terms of y :
x − y = 1 x = y + 1
Substitute x in the first equation Next, we substitute the expression for x into the first equation:
2 x + y = 8 2 ( y + 1 ) + y = 8
Solve for y Now, we simplify and solve for y :
2 ( y + 1 ) + y = 8 2 y + 2 + y = 8 3 y + 2 = 8 3 y = 6 y = 2
Solve for x Now that we have the value of y , we substitute it back into the equation x = y + 1 to find the value of x :
x = y + 1 x = 2 + 1 x = 3
State the solution Therefore, the solution to the system of equations is x = 3 and y = 2 . We can write this as the ordered pair ( 3 , 2 ) .
Examples
Substitution method is a powerful tool in various real-life scenarios. For instance, consider a scenario where you're trying to optimize your study time between two subjects, Math and Science. Let's say you have a total of 10 hours to study ( M + S = 10 ). You also know that for every hour you spend on Math, you want to spend twice as much time on Science ( S = 2 M ). Using substitution, you can solve this system of equations to determine the optimal number of hours to dedicate to each subject, ensuring you balance your study efforts effectively. This method is also applicable in finance, resource allocation, and many other fields where interrelated variables need to be determined.
