How many solutions does the equation y = 3x + 5 have? Write four solutions for the equation 2x + y = 7 and pi x + y = 9.

The equation $y = 3x + 5$ is a linear equation in two variables ($x$ and $y$).

A linear equation in two variables represents a straight line in a coordinate plane. Every point on this line is a solution to the equation. Since a straight line consists of an infinite number of points, a linear equation in two variables has infinitely many solutions.

Therefore, option (iii) is true.
The equation $y = 3x + 5$ has (iii) infinitely many solutions.

2.
We need to write four solutions for each equation.

• 2(i) $2x + y = 7$

• If $x = 0$, $y = 7 - 2(0) = 7$. Solution: $(0, 7)$
• If $x = 1$, $y = 7 - 2(1) = 5$. Solution: $(1, 5)$
• If $x = 2$, $y = 7 - 2(2) = 3$. Solution: $(2, 3)$
• If $x = 3$, $y = 7 - 2(3) = 1$. Solution: $(3, 1)$ Four solutions are: $\boxed{(0, 7), (1, 5), (2, 3), (3, 1)}$

• 2(ii) $\pi x + y = 9$

• If $x = 0$, $y = 9 - \pi(0) = 9$. Solution: $(0, 9)$
• If $x = 1$, $y = 9 - \pi(1) = 9 - \pi$. Solution: $(1, 9 - \pi)$
• If $x = 2$, $y = 9 - \pi(2) = 9 - 2\pi$. Solution: $(2, 9 - 2\pi)$
• If $x = -1$, $y = 9 - \pi(-1) = 9 + \pi$. Solution: $(-1, 9 + \pi)$ Four solutions are: $\boxed{(0, 9), (1, 9 - \pi), (2, 9 - 2\pi), (-1, 9 + \pi)}$

• 2(iii) $x = 4y$

(Already done) $x = 4y$

• If $y = 0$, $x = 4(0) = 0$. Solution: $(0, 0)$
• If $y = 1$, $x = 4(1) = 4$. Solution: $(4, 1)$
• If $y = 2$, $x = 4(2) = 8$. Solution: $(8, 2)$
• If $y = -1$, $x = 4(-1) = -4$. Solution: $(-4, -1)$ Four solutions are: $\boxed{(0, 0), (4, 1), (8, 2), (-4, -1)}$

3.
We need to check which of the given points are solutions to the equation $x - 2y = 4$.

• 3(i) $(0, 2)$

Since $-4 \neq 4$, $(0, 2)$ is **not a solution**.

• 3(ii) $(2, 0)$

Since $2 \neq 4$, $(2, 0)$ is **not a solution**.

• 3(iii) $(4, 0)$

Since $4 = 4$, $(4, 0)$ **is a solution**.

• 3(iv) $(\sqrt{2}, 4\sqrt{2})$

Since $-7\sqrt{2} \neq 4$, $(\sqrt{2}, 4\sqrt{2})$ is **not a solution**.

• 3(v) $(1, 1)$

Since $-1 \neq 4$, $(1, 1)$ is **not a solution**.

Here's a summary of the solutions for $x - 2y = 4$:

• $(0, 2)$ is not a solution.
• $(2, 0)$ is not a solution.
• $(4, 0)$ is a solution.
• $(\sqrt{2}, 4\sqrt{2})$ is not a solution.
• $(1, 1)$ is not a solution.