Utility function when goods are perfect substitutes

Summary

• Perfect substitutes with a one-to-one substitution: $u\left( x_1,x_2 \right)=x_1+x_2$
• Any monotonic transformation such as $4x_1+4x_2$ or ${\left( x_1+x_2 \right)}^2$ will do as well.
• Perfect substitutes with an arbitrary rate of substitution: $u\left( x_1,x_2 \right)=ax_1+bx_2$
• $MRS=-a/b$