For what value of [tex]$k$[/tex] do the equations [tex]$3x - 2y = 5$[/tex] and [tex]$x + y = k + 2$[/tex] have [tex]$P_x = 15$[/tex]?
Answer (2)
To find the value of k , substitute x = 15 into the first equation, leading to y = 20 . Then, by substituting these values into the second equation, we find that k = 33 . Thus, the value of k is 33 .
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Substitute x = 15 into the equation 3 x − 2 y = 5 and solve for y , which gives y = 20 .
Substitute x = 15 and y = 20 into the equation x + y = k + 2 .
Simplify the equation to find 35 = k + 2 .
Solve for k , which gives k = 33 . The final answer is 33 .
Explanation
Understanding the Problem We are given two equations: 3 x − 2 y = 5 and x + y = k + 2 . We are also given that P x = 15 , which means x = 15 . Our goal is to find the value of k .
Solving for y First, substitute x = 15 into the first equation to solve for y :
3 ( 15 ) − 2 y = 5 45 − 2 y = 5 − 2 y = 5 − 45 − 2 y = − 40 y = − 2 − 40 y = 20
Solving for k Now, substitute x = 15 and y = 20 into the second equation to solve for k :
x + y = k + 2 15 + 20 = k + 2 35 = k + 2 k = 35 − 2 k = 33
Final Answer Therefore, the value of k is 33.
Examples
Understanding systems of equations is crucial in various real-world applications. For instance, consider a scenario where you're trying to optimize a budget. You have two investment options, each with different rates of return and risk levels. By setting up a system of equations, you can determine the optimal amount to invest in each option to achieve your financial goals while staying within your risk tolerance. This involves solving for the unknowns (investment amounts) based on the given constraints (total budget, desired return).
