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 (1)
Substitute x = 15 into the first equation and solve for y : 3 ( 15 ) − 2 y = 5 A rry = 20 .
Substitute x = 15 and y = 20 into the second equation: 15 + 20 = k + 2 .
Simplify the equation: 35 = k + 2 .
Solve for k : k = 33 , so 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 the x-coordinate of the solution to the system of equations is 15. Our goal is to find the value of k .
Substituting x in the First Equation First, substitute x = 15 into the first equation to solve for y :
3 ( 15 ) − 2 y = 5
Simplifying Simplify the equation: 45 − 2 y = 5
Isolating the y term Subtract 45 from both sides: − 2 y = 5 − 45
Simplifying − 2 y = − 40
Solving for y Divide both sides by -2: y = − 2 − 40 y = 20
Substituting x and y in the Second Equation Now that we have x = 15 and y = 20 , substitute these values into the second equation: x + y = k + 2 15 + 20 = k + 2
Simplifying Simplify the equation: 35 = k + 2
Solving for k Subtract 2 from both sides to solve for k :
k = 35 − 2 k = 33
Final Answer Therefore, the value of k is 33.
Examples
Imagine you're designing a seesaw where the balance depends on the weights and positions of two people. This problem is similar to finding the right balance (k) in a system of equations, given one person's position (x). By solving the equations, you ensure the seesaw is perfectly balanced, just like finding the correct value of k ensures the equations are consistent.
