Let's solve each of the problems one by one by finding the equations of the lines in standard form, $Ax + By = C$, with the given properties.

1. Slope = 3, y-intercept = 1

The slope-intercept form is: $y = mx + b$ Here, the slope $m = 3$ and the y-intercept $b = 1$, so: $y = 3x + 1$ Convert to standard form: $y - 3x = 1 \quad \Rightarrow \quad 3x - y = -1$ Thus, the equation is: $3x - y = -1$

2. Passing through (0,2), slope = -4

The slope-intercept form is: $y - y_1 = m(x - x_1)$ Using $(x_1, y_1) = (0, 2)$ and slope $m = -4$: $y - 2 = -4(x - 0)$ $y - 2 = -4x \quad \Rightarrow \quad y = -4x + 2$ Convert to standard form: $4x + y = 2$

3. Passing through (-1,3) and (1,1)

First, find the slope: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 3}{1 - (-1)} = \frac{-2}{2} = -1$ Using the point-slope form with point $(-1, 3)$ and slope $m = -1$: $y - 3 = -1(x + 1)$ $y - 3 = -x - 1 \quad \Rightarrow \quad y = -x + 2$ Convert to standard form: $x + y = 2$

4. Passing through (1,3), slope = 1/2

Using point-slope form with point $(1, 3)$ and slope $m = \frac{1}{2}$: $y - 3 = \frac{1}{2}(x - 1)$ $y - 3 = \frac{1}{2}x - \frac{1}{2} \quad \Rightarrow \quad y = \frac{1}{2}x + \frac{5}{2}$ Convert to standard form: $2y = x + 5 \quad \Rightarrow \quad x - 2y = -5$

5. Passing through (1/2, 1) and (4, 2)

First, find the slope: $m = \frac{2 - 1}{4 - \frac{1}{2}} = \frac{1}{\frac{7}{2}} = \frac{2}{7}$ Using point-slope form with point $(\frac{1}{2}, 1)$ and slope $m = \frac{2}{7}$: $y - 1 = \frac{2}{7}\left(x - \frac{1}{2}\right)$ $y - 1 = \frac{2}{7}x - \frac{1}{7} \quad \Rightarrow \quad y = \frac{2}{7}x + \frac{6}{7}$ Convert to standard form: $7y = 2x + 6 \quad \Rightarrow \quad 2x - 7y = -6$

Summary of the Standard Forms:

1. $3x - y = -1$
2. $4x + y = 2$
3. $x + y = 2$
4. $x - 2y = -5$
5. $2x - 7y = -6$

Would you like further details or explanations on any of these steps?

Here are 5 related questions:

1. How do you convert slope-intercept form to standard form?
2. What is the formula for finding the slope between two points?
3. Can a line have a slope of 0? What would that look like?
4. How would you write the equation of a vertical line?
5. What happens if two points have the same x-value? How does that affect the slope?

Tip: In standard form, it’s common practice to make sure the coefficient of $x$ is positive and all coefficients are integers.