To determine the equation of a line in the form $y = mx + c$, we need to find the slope $m$ and the y-intercept $c$. Here's how we calculate them:

1. Slope $m$: The slope of the line is the change in $y$ divided by the change in $x$. If you have two points on the line, say $(x_1, y_1)$ and $(x_2, y_2)$, the slope is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

2. Y-intercept $c$: The y-intercept is the point where the line crosses the y-axis. This occurs when $x = 0$. To find $c$, use the equation of the line with a known point $(x_1, y_1)$ and the calculated slope: $y_1 = m x_1 + c$ Solve for $c$: $c = y_1 - m x_1$

Once you know $m$ and $c$, you can write the full equation of the line.

Example:

Suppose you're given two points: $(1, 2)$ and $(3, 6)$.

• The slope is: $m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2$

• To find the y-intercept, use one of the points, say $(1, 2)$, in the equation: $2 = 2(1) + c \quad \Rightarrow \quad c = 0$

Thus, the equation of the line is: $y = 2x$

Would you like more details, or do you have any questions?

Related Questions:

1. How do you find the slope of a vertical line?
2. What happens when the slope $m$ is zero?
3. How do you calculate the slope if the points are fractions?
4. Can you determine the equation of a line given only one point and the slope?
5. How can you verify if a point lies on a given line?

Tip:

To quickly determine the slope, remember it represents how steep the line is—higher values of $m$ mean a steeper line!