The Slope of the Line y=mx+b

By visually determining the rise and run from the graph you can notice that the rise (change in $y$) is up $4$ units, and the run (change in $x$) is right $3$ units. The slope of the line through $(-4, -2)$ and $(-1, 2)$ is $\frac{4}{3}$, indicating a positive slope.

For the points $(2, 9)$ and $(5, 3)$, the rise (or in this case, the fall) is $3 - 8 = -5$ and the run is $5 - 2 = 3$. Therefore, the slope of the line is $\frac{-5}{3}$, indicating a negative slope as the line falls $\frac{-5}{3}$ units for every $1$ unit it moves to the right.

Looking at the points $(1, -5)$ and $(4, -1)$, the rise is $-1 - (-5) = 4$ and the run is $4 - 1 = 3$. Thus, the slope of the line is $\frac{4}{3}$, demonstrating a positive slope as the line ascends $4$ units vertically for every $3$ units it progresses horizontally.

To calculate the slope, use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Plugging in the given points, we get $m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$. Therefore, the slope of the line is $3$, which is a positive slope.

Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, the slope is $m = \frac{-5 - (-1)}{2 - 4} = \frac{-4}{-2} = 2$. So, the slope of the line is $2$.

Applying the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ to the points $(-2, 3)$ and $(-2, -4)$, we find the denominator is $-2 - (-2) = 0$. Since division by zero is undefined, the slope of this line is undefined, indicating a vertical line.

To calculate the slope, use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. For these points, $m = \frac{0 - 4}{3 - (-1)} = \frac{-4}{4} = -1$. Therefore, the slope of the line is $-1$, indicating a negative slope.

Using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, the slope is calculated as $m = \frac{-1 - 3}{5 - 2} = \frac{-4}{3}.Thus, the slope of this line has a negative slope.

To find the slope, use $m = \frac{y_2 - y_1}{x_2 - x_1}$. Substituting the given points gives us $m = \frac{15 - 5}{10 - 5} = \frac{10}{5} = 2$. Thus, the slope of the line is $2$.

The slope of the line is $-3$. This is because in the slope-intercept form $y = mx + c$, the coefficient of $x$ ($m$) represents the slope, which is $-3$ in this equation.

The slope of the line is $\frac{1}{2}$. In the equation $y = mx + c$, the slope is the coefficient of $x$, which is $\frac{1}{2}$ here.

The slope of the line is $5$. In the equation $y = mx$, the slope is the coefficient of $x$, which is $5$ in this case, indicating a steep ascent.

The slope of the line is $-\frac{4}{5}$. The coefficient of $x$ in the equation $y = mx + c$ gives the slope, which is $-\frac{4}{5}$, showing a moderate decline to the right.

The rise from $-2$ to $4$ is $6$ units, and the run from $0$ to $3$ is $3$ units. Therefore, the slope of the line is $\frac{6}{3} = 2$.

The slope of the line is $-\frac{3}{2}$. This is identified by the coefficient of $x$ in the equation $y = mx + c$, which is $-\frac{3}{2}$, indicating the line descends $\frac{3}{2}$ units for every 1 unit it moves to the right.

The rise is $-3 - 3 = -6$ units, and the run is $-2 - (-5) = 3$ units. Thus, the slope of the line is $\frac{-6}{3} = -2$, showing a negative slope.

The slope of the line is $3$. In the slope-intercept form $y = mx + c$, the coefficient of $x$, which is $3$, represents the slope, indicating a steep ascent to the right.

Applying the slope formula, the rise is $-6 - 2 = -8$ units, and the run is $10 - 6 = 4$ units. Therefore, the slope is $\frac{-8}{4} = -2$.

The slope of the line is $\frac{2}{3}$. The coefficient of $x$ in the equation, $\frac{2}{3}$, directly indicates the slope, demonstrating that for every 1 unit increase in $x$, $y$ increases by $\frac{2}{3}$ of a unit.

The slope is a measure that tells us how steep a line is. It's calculated as the ratio of the vertical change (rise) over the horizontal change (run) between two points on the line, represented mathematically as $m = \frac{\Delta y}{\Delta x}$.

To calculate the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$, use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. This formula gives the slope 'm' of the line passing through these points.

A positive slope indicates that the line rises to the right. Graphically, as you move from left to right, the line goes upward. This shows that the $y$-value (or the vertical position) increases as the $x$-value (or the horizontal position) increases.

A negative slope indicates that the line falls to the right. This means as you move from left to right along the line, the $y$-value decreases. It shows a decrease in the vertical position as the horizontal position increases.

A line with a zero slope is perfectly horizontal. This means there's no vertical change as you move along the line, represented as $m = 0$. It signifies no increase or decrease in the $y$-value regardless of the $x$-value's changes.

An undefined slope occurs with vertical lines, where there's no horizontal change as you move along the line. Since you cannot divide by zero, the slope formula $m = \frac{\Delta y}{\Delta x}$ becomes undefined due to $\Delta x = 0$.

Yes, the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ can be applied to any line except for vertical lines, which have an undefined slope. For vertical lines, the change in $x$ is zero, making the denominator of the slope formula zero.

In the slope-intercept form of a line, $y = mx + c$, the slope 'm' determines how steep the line is and whether it slopes up or down. The $y$-intercept 'c' indicates where the line crosses the $y$-axis.

Two lines are parallel if and only if they have the same slope. This means they rise or fall at the same rate and will never intersect, no matter how far they extend.

Perpendicular lines have slopes that are negative reciprocals of each other. This means if one line has a slope of $m$, the perpendicular line will have a slope of $-\frac{1}{m}$. For example, if one line's slope is $\frac{3}{4}$, the perpendicular line's slope would be $-\frac{4}{3}$.