1. Which of the following equations represent(s) a line that goes through the points $$ (2,3) $$ and $$ (1,1) $$ ? I) $$ y=2(x-2)+3 $$ II) $$ y=2 x-1 $$ III) $$ y=2(x-1)+1 $$ II and III III only

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To determine which of the given equations represent a line that goes through the points $$ (2,3) $$ and $$ (1,1) $$, we first need to find the slope of the line that passes through these two points.

### Step 1: Calculate the slope

The slope $$ m $$ of a line through two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by the formula:

$$
m = \frac{y_2 - y_1}{x_2 - x_1}
$$

Using the points $$ (2,3) $$ and $$ (1,1) $$:

- $$ (x_1, y_1) = (2, 3) $$
- $$ (x_2, y_2) = (1, 1) $$

Substituting these values into the slope formula:

$$
m = \frac{1 - 3}{1 - 2} = \frac{-2}{-1} = 2
$$

### Step 2: Find the equation of the line

Now that we have the slope $$ m = 2 $$, we can use the point-slope form of the equation of a line, which is:

$$
y - y_1 = m(x - x_1)
$$

Using point $$ (2, 3) $$:

$$
y - 3 = 2(x - 2)
$$

Expanding this equation:

$$
y - 3 = 2x - 4
$$
$$
y = 2x - 1
$$

### Step 3: Check the given equations

Now we will check each of the given equations to see if they match the derived equation or if they pass through the points $$ (2,3) $$ and $$ (1,1) $$.

1. **Equation I: $$ y = 2(x - 2) + 3 $$**

Expanding this:
   $$
   y = 2x - 4 + 3 = 2x - 1
   $$
   This matches our derived equation.

2. **Equation II: $$ y = 2x - 1 $$**

This is exactly the same as our derived equation.

3. **Equation III: $$ y = 2(x - 1) + 1 $$**

Expanding this:
   $$
   y = 2x - 2 + 1 = 2x - 1
   $$
   This also matches our derived equation.

### Conclusion

All three equations represent the same line that goes through the points $$ (2,3) $$ and $$ (1,1) $$. Therefore, the correct answer is that all three equations represent a line through the given points.

Thus, the answer is **none of the options provided (I, II, III)**.
All three equations represent a line that goes through the points $$ (2,3) $$ and $$ (1,1) $$.

1. Which of the following equations represent(s) a line that goes through the points \( (2,3) \) and \( (1,1) \) ? I) \( y=2(x-2)+3 \) II) \( y=2 x-1 \) III) \( y=2(x-1)+1 \) II and III III only

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