1. **Problem:** Show that if the linear equations $x_1 + kx_2 = c$ and $x_1 + lx_2 = d$ have the same solution set, then the two equations are identical (i.e., $k = l$ and $c = d$). 2. **Formula and rules:** Two linear equations represent the same line if and only if one is a scalar multiple of the other. Since both equations have the same $x_1$ coefficient (which is 1), the only way for them to be identical is if their other coefficients and constants are equal. 3. **Work:** - Given the equations: $$x_1 + kx_2 = c$$ $$x_1 + lx_2 = d$$ - Suppose they have the same solution set. - Subtract the second equation from the first: $$ (x_1 + kx_2) - (x_1 + lx_2) = c - d $$ $$ (k - l)x_2 = c - d $$ - For this to hold for all $x_2$, the coefficient of $x_2$ and the constant must be zero: $$ k - l = 0 \implies k = l $$ $$ c - d = 0 \implies c = d $$ 4. **Explanation:** Since the difference of the two equations must be zero for all $x_2$, the coefficients and constants must be equal, proving the equations are identical. **Final answer:** $k = l$ and $c = d$