1. **State the problem:** Solve the system of equations: $$y^2 = 3x^2 + 4$$ $$y + 2x = 7$$ for $x$ and $y$. 2. **Use substitution:** From the linear equation, express $y$ in terms of $x$: $$y = 7 - 2x$$ 3. **Substitute into the quadratic equation:** Replace $y$ in the first equation: $$ (7 - 2x)^2 = 3x^2 + 4 $$ 4. **Expand and simplify:** $$ 49 - 28x + 4x^2 = 3x^2 + 4 $$ Bring all terms to one side: $$ 4x^2 - 3x^2 - 28x + 49 - 4 = 0 $$ $$ x^2 - 28x + 45 = 0 $$ 5. **Solve the quadratic equation:** Use the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=1$, $b=-28$, $c=45$. Calculate the discriminant: $$ \Delta = (-28)^2 - 4 \times 1 \times 45 = 784 - 180 = 604 $$ 6. **Calculate the roots:** $$ x = \frac{28 \pm \sqrt{604}}{2} $$ Approximate $\sqrt{604} \approx 24.6$: $$ x_1 = \frac{28 + 24.6}{2} = \frac{52.6}{2} = 26.3 $$ $$ x_2 = \frac{28 - 24.6}{2} = \frac{3.4}{2} = 1.7 $$ 7. **Find corresponding $y$ values:** For $x_1 = 26.3$: $$ y_1 = 7 - 2 \times 26.3 = 7 - 52.6 = -45.6 $$ For $x_2 = 1.7$: $$ y_2 = 7 - 2 \times 1.7 = 7 - 3.4 = 3.6 $$ 8. **Final answers rounded to three significant figures:** $$ x = 26.3, y = -45.6 $$ $$ x = 1.70, y = 3.60 $$