To solve the given system of equations:

1. $y = 2x - 3$
2. $x^2 + y^2 = 2$

Steps:

1. Substitution:

From the first equation, substitute $y = 2x - 3$ into the second equation: $x^2 + (2x - 3)^2 = 2$

2. Expand $(2x - 3)^2$:

$(2x - 3)^2 = 4x^2 - 12x + 9$ Substitute this back: $x^2 + 4x^2 - 12x + 9 = 2$

3. Simplify:

$5x^2 - 12x + 9 = 2$ $5x^2 - 12x + 7 = 0$

4. Solve the quadratic equation:

Using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 5$, $b = -12$, and $c = 7$: $x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(5)(7)}}{2(5)}$ $x = \frac{12 \pm \sqrt{144 - 140}}{10}$ $x = \frac{12 \pm \sqrt{4}}{10}$ $x = \frac{12 \pm 2}{10}$

5. Solve for $x$:

$x = \frac{12 + 2}{10} = \frac{14}{10} = 1.4$ $x = \frac{12 - 2}{10} = \frac{10}{10} = 1$

6. Find $y$ for each $x$:

• For $x = 1.4$: $y = 2(1.4) - 3 = 2.8 - 3 = -0.2$

• For $x = 1$: $y = 2(1) - 3 = 2 - 3 = -1$

Solutions:

The solutions are: $(x, y) = (1.4, -0.2) \quad \text{and} \quad (x, y) = (1, -1)$

Would you like a detailed graph or explanation of any step?

Related Questions:

1. How do you verify the solutions graphically?
2. How does substitution work in solving simultaneous equations?
3. What does the quadratic formula represent geometrically?
4. How do you check if a solution satisfies both equations?
5. What are the possible scenarios when solving simultaneous equations (e.g., no solutions, one solution, or infinite solutions)?

Tip:

Always double-check your solutions by substituting them back into both equations.