If \(2x + y = 23\) and \(4x - y = 19\), find the values of \(5y - 2x\) and \(\dfrac{y}{x} - 2\).
\(5y - 2x = 31\) and \(\dfrac{y}{x} - 2 = -\dfrac{5}{7}\).
Step 1: Write the two equations clearly.
Equation (1): \(2x + y = 23\)
Equation (2): \(4x - y = 19\)
Step 2: Add the two equations to remove \(y\).
\((2x + y) + (4x - y) = 23 + 19\)
\(2x + 4x + y - y = 42\)
\(6x = 42\)
So, \(x = 7\).
Step 3: Put the value of \(x\) into Equation (1).
Equation (1): \(2x + y = 23\)
\(2(7) + y = 23\)
\(14 + y = 23\)
So, \(y = 23 - 14 = 9\).
Step 4: Now calculate \(5y - 2x\).
\(5y - 2x = 5(9) - 2(7)\)
\(= 45 - 14\)
\(= 31\)
Step 5: Calculate \(\dfrac{y}{x} - 2\).
\(\dfrac{y}{x} - 2 = \dfrac{9}{7} - 2\)
Write 2 as \(\dfrac{14}{7}\).
So, \(\dfrac{9}{7} - \dfrac{14}{7} = \dfrac{-5}{7}\).
Final Answer:
\(5y - 2x = 31\)
\(\dfrac{y}{x} - 2 = -\dfrac{5}{7}\)