In the rectangle, opposite sides are equal. Given the labels in Fig. 3.2, find \(x\) and \(y\): top \(= x + 3y\), bottom \(= 13\), left \(= 3x + y\), right \(= 7\).
\(x = 1\) and \(y = 4\).
In a rectangle, opposite sides are equal in length.
So we can write two equations:
Top = Bottom → \(x + 3y = 13\)
Left = Right → \(3x + y = 7\)
Step 1: Start with the second equation.
\(3x + y = 7\)
Subtract \(3x\) from both sides:
\(y = 7 - 3x\)
Step 2: Put this value of \(y\) into the first equation.
The first equation is:
\(x + 3y = 13\)
Replace \(y\) with \(7 - 3x\):
\(x + 3(7 - 3x) = 13\)
Step 3: Expand the brackets:
\(x + 21 - 9x = 13\)
Step 4: Combine like terms:
\(-8x + 21 = 13\)
Step 5: Subtract 21 from both sides:
\(-8x = -8\)
Step 6: Divide both sides by \(-8\):
\(x = 1\)
Step 7: Now find \(y\) using \(y = 7 - 3x\):
\(y = 7 - 3(1)\)
\(y = 7 - 3 = 4\)
So the solution is:
\(x = 1\) and \(y = 4\).