What is the value of y in the solution to the following system of linear equations?
[tex]2x+2=y[/tex]
[tex]2y=5x-1[/tex]
Answer (2)
Solve the first equation for x : x = 2 y − 2 .
Substitute x into the second equation: 2 y = 5 ( 2 y − 2 ) − 1 .
Simplify and solve for y : y = 12 .
The value of y is 12 .
Explanation
Analyze the problem We are given a system of two linear equations:
2 x + 2 = y 2 y = 5 x − 1
Our goal is to find the value of y that satisfies both equations.
Solve for x First, we can solve the first equation for x in terms of y :
2 x = y − 2 x = 2 y − 2
Substitute x into the second equation Next, we substitute this expression for x into the second equation:
2 y = 5 ( 2 y − 2 ) − 1
Solve for y Now, we simplify the second equation and solve for y :
2 y = 2 5 y − 10 − 1 Multiply both sides by 2 to eliminate the fraction:
4 y = 5 y − 10 − 2 4 y = 5 y − 12 Subtract 5 y from both sides:
− y = − 12 Multiply both sides by -1:
y = 12
Final Answer Therefore, the value of y in the solution to the system of equations is 12.
Examples
Systems of equations are incredibly useful in real-world scenarios. For example, imagine you're running a small business selling two types of products. You know the total revenue you made from selling a certain quantity of both products, and you also know the cost of producing each product. By setting up a system of equations, you can determine exactly how many of each product you sold and optimize your production to maximize profit. This kind of problem-solving is fundamental in economics, business management, and many other fields.
The value of y in the solution to the given system of equations is 12. We found this by substituting and rearranging the equations step by step. Ultimately, solving the simplified equation led us to this result.
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