1. **State the problem:** We have two rectangles H I J K and L M N O that are equivalent, meaning they have the same area. We need to find the length of side L M. 2. **Write down the given dimensions:** - Rectangle H I J K has height $x+1$ cm and width $4x-3$ cm. - Rectangle L M N O has height $x$ cm and width $4x$ cm. 3. **Set up the equation for equivalent areas:** $$\text{Area of H I J K} = \text{Area of L M N O}$$ $$ (x+1)(4x-3) = x \times 4x $$ 4. **Expand and simplify:** $$ (x+1)(4x-3) = 4x^2 $$ $$ 4x^2 - 3x + 4x - 3 = 4x^2 $$ $$ 4x^2 + x - 3 = 4x^2 $$ 5. **Subtract $4x^2$ from both sides:** $$ \cancel{4x^2} + x - 3 = \cancel{4x^2} $$ $$ x - 3 = 0 $$ 6. **Solve for $x$:** $$ x = 3 $$ 7. **Find the length of $L M$ which is $4x$ cm:** $$ L M = 4x = 4 \times 3 = 12 \text{ cm} $$ **Final answer:** The length of $L M$ is 12 cm.