1. **Problem:** Solve the simultaneous equations: y + x = 3 and y^2 - x^2 = 5. 2. **Formula and rules:** - From the first equation, express y in terms of x: $$y = 3 - x$$. - Substitute into the second equation: $$y^2 - x^2 = 5$$. - Recall the identity: $$a^2 - b^2 = (a - b)(a + b)$$. 3. **Substitution and simplification:** Substitute $$y = 3 - x$$ into $$y^2 - x^2 = 5$$: $$ (3 - x)^2 - x^2 = 5 $$ Expand $$ (3 - x)^2 $$: $$ 9 - 6x + x^2 - x^2 = 5 $$ Simplify: $$ 9 - 6x = 5 $$ 4. **Solve for x:** $$ 9 - 6x = 5 $$ $$ -6x = 5 - 9 $$ $$ -6x = -4 $$ $$ x = \frac{-4}{-6} = \frac{2}{3} $$ 5. **Find y:** Using $$ y = 3 - x $$: $$ y = 3 - \frac{2}{3} = \frac{9}{3} - \frac{2}{3} = \frac{7}{3} $$ 6. **Final answer:** $$ x = \frac{2}{3}, \quad y = \frac{7}{3} $$ This corresponds to approximately $$ x = 0.67, y = 2.33 $$. **Note:** The closest option given is x = 1½, y = 2½, but the exact solution is as above.