How do you determine if a point is a solution to a system of equations, what does it mean if a system has no solution, and what are the two numbers if their sum is 25 and the larger is one more than t

Given the system of equations:

1. $3x - 2y - 9 = 0$
2. $x + 8y - 3 = 0$ To check if $(3, 0)$ is a solution, substitute $x=3$ and $y=0$ into each equation.

For equation 1:

$$3(3) - 2(0) - 9 = 9 - 0 - 9 = 0$$

This equation holds true.

For equation 2:

$$3 + 8(0) - 3 = 3 + 0 - 3 = 0$$

This equation also holds true.
Since $(3, 0)$ satisfies both equations, it is a solution.

Answer: Yes

If a system of equations has no solution, it means the lines represented by the equations never intersect. This occurs when the lines are parallel and distinct.

Answer: parallel

Let the smaller number be $s$ and the larger number be $l$.
From the problem statement, we have two conditions:

1. The sum of two numbers is 25:

$$s + l = 25$$

2. The larger number is one more than twice the smaller number:

$$l = 2s + 1$$

Substitute the expression for $l$ from the second equation into the first equation:

$$s + (2s + 1) = 25$$ $$3s + 1 = 25$$

Subtract 1 from both sides:

$$3s = 24$$

Divide by 3:

$$s = 8$$

Now substitute the value of $s$ back into the equation for $l$:

$$l = 2(8) + 1$$ $$l = 16 + 1$$ $$l = 17$$

The larger number is 17.

Answer: $\boxed{17}$

Given the system of equations:

1. $3a + 5b = 7$
2. $a - 2b = 4$ From equation (2), express $a$ in terms of $b$:

$$a = 4 + 2b$$

Substitute this expression for $a$ into equation (1):

$$3(4 + 2b) + 5b = 7$$ $$12 + 6b + 5b = 7$$ $$12 + 11b = 7$$

Subtract 12 from both sides:

$$11b = 7 - 12$$ $$11b = -5$$ $$b = -\frac{5}{11}$$

Now substitute the value of $b$ back into the expression for $a$:

$$a = 4 + 2\left(-\frac{5}{11}\right)$$ $$a = 4 - \frac{10}{11}$$

To subtract, find a common denominator:

$$a = \frac{4 \times 11}{11} - \frac{10}{11}$$ $$a = \frac{44}{11} - \frac{10}{11}$$ $$a = \frac{34}{11}$$

The solution $(a, b)$ is $\left(\frac{34}{11}, -\frac{5}{11}\right)$.

Answer: $\boxed{\left(\frac{34}{11}, -\frac{5}{11}\right)}$