2a rectangle's width is three times its height, and its perimiter is 120 mm . Find the rectangle's width and its height.

Let's denote the width of the rectangle as \( w \) and the height as \( h \). Given: 1. The width is three times the height: \( w = 3h \) 2. The perimeter of the rectangle is 120 mm: \( 2w + 2h = 120 \) We can solve these equations to find the width and height of the rectangle. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}w=3h\\2w+2h=120\end{array}\right.\) - step1: Substitute the value of \(w:\) \(2\times 3h+2h=120\) - step2: Simplify: \(8h=120\) - step3: Divide both sides: \(\frac{8h}{8}=\frac{120}{8}\) - step4: Divide the numbers: \(h=15\) - step5: Substitute the value of \(h:\) \(w=3\times 15\) - step6: Simplify: \(w=45\) - step7: Calculate: \(\left\{ \begin{array}{l}h=15\\w=45\end{array}\right.\) - step8: Check the solution: \(\left\{ \begin{array}{l}h=15\\w=45\end{array}\right.\) - step9: Rewrite: \(\left(h,w\right) = \left(15,45\right)\) The solution to the system of equations is \( h = 15 \) and \( w = 45 \). Therefore, the width of the rectangle is 45 mm and the height is 15 mm.