4. Graphing Linear Equations in Two Variables

Slope of a Line

4. Graphing Linear Equations in Two Variables

Slope of a Line: Videos & Practice Problems

Topic summary

The slope of a line, represented by $m$ , measures its steepness and is calculated as the rise over run, or $\frac{Δy}{Δx}$ . Slopes can be positive, negative, zero, or undefined. Positive slopes rise left to right, negative slopes fall, zero slope indicates a horizontal line ( $y = constant$ ), and undefined slope corresponds to vertical lines ( $x = constant$ ). Understanding slope is essential for analyzing linear equations and graph behavior in algebra and coordinate geometry.

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Finding the Slope of a Line

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Finding the Slope of a Line Video Summary

The slope of a line is a fundamental concept in understanding the characteristics of linear equations. Represented by the letter m, the slope measures the steepness or incline of a line. It is calculated by dividing the vertical change, known as the rise, by the horizontal change, called the run. This relationship can be expressed simply as slope = rise/run.

To find the slope from a graph, identify two points on the line and determine how much the line rises vertically and runs horizontally between these points. For example, if the line rises 3 units and runs 1 unit to the right, the slope is calculated as $m = \frac{3}{1} = 3$. This indicates a relatively steep line.

More formally, the slope can be calculated using the coordinates of two points, $(x_1, y_1)$ and $(x_2, y_2)$, on the line. The formula for slope is:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

This formula represents the change in the y-values (vertical change) divided by the change in the x-values (horizontal change), often referred to as $\Delta y$ over $\Delta x$, where the Greek letter delta ($\Delta$) signifies "change in". For instance, if the points are $(-1, 0)$ and $(0, 3)$, the slope calculation would be:

\[m = \frac{3 - 0}{0 - (-1)} = \frac{3}{1} = 3\]

It is important to note that the order in which you subtract the coordinates must be consistent; however, it does not matter which point is chosen as $(x_1, y_1)$ or $(x_2, y_2)$ as long as the subtraction order is maintained.

When given only two points without a graph, the same formula applies. For example, with points $(1, 2)$ and $(4, 5)$, the slope is:

\[m = \frac{5 - 2}{4 - 1} = \frac{3}{3} = 1\]

This slope of 1 indicates a less steep line compared to a slope of 3, reflecting the different inclines of the lines represented by these points.

Understanding how to calculate slope is essential for analyzing linear relationships, graphing lines, and solving problems involving rates of change. Mastery of this concept enables one to interpret the behavior of lines in coordinate geometry effectively.

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Finding the Slope of a Line Example 1

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Finding the Slope of a Line Example 1 Video Summary

To find the slope of a line given two points, it is essential to understand that the slope represents the ratio of the vertical change (rise) to the horizontal change (run) between those points. The slope formula is expressed as the difference in the y-coordinates divided by the difference in the x-coordinates, written mathematically as:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points on the line. The order of the points does not affect the slope calculation as long as the subtraction is consistent.

For example, consider the points $(-2, -5)$ and $(4, 3)$. Assigning $x_1 = -2$, $y_1 = -5$, $x_2 = 4$, and $y_2 = 3$, substitute these values into the slope formula:

\[ m = \frac{3 - (-5)}{4 - (-2)} = \frac{3 + 5}{4 + 2} = \frac{8}{6} \]

Simplifying the fraction by dividing numerator and denominator by their greatest common divisor, 2, yields:

\[ m = \frac{4}{3} \]

This means the line rises 4 units vertically for every 3 units it runs horizontally. To verify this slope, one can select any point on the line and check that moving 4 units up and 3 units to the right lands on another point on the line, confirming the slope's accuracy.

Understanding how to calculate slope is fundamental in coordinate geometry, as it describes the steepness and direction of a line. Mastery of this concept enables solving various problems involving linear relationships and graph interpretation.

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Finding the Slope of a Line Example 2

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Finding the Slope of a Line Example 2 Video Summary

To determine the slope of a line given two points, we use the fundamental concept that slope represents the ratio of the vertical change (rise) to the horizontal change (run) between those points. Mathematically, the slope $ m $ is calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points on the line. The order of points does not affect the slope calculation as long as the subtraction is consistent.

For example, consider the points $(-4, 4)$ and $(4, 0)$. Applying the slope formula:

\[ m = \frac{0 - 4}{4 - (-4)} = \frac{-4}{8} = -\frac{1}{2} \]

This result means the line decreases by 1 unit vertically for every 2 units it moves horizontally to the right, indicating a negative slope of $-\frac{1}{2}$.

To verify this slope, you can use the rise-over-run method visually on the graph. Starting from a point on the line, move down 1 unit (rise of $-1$) and then move 2 units to the right (run of 2). If this movement lands on another point on the line, the slope calculation is confirmed.

Understanding slope is essential in algebra and coordinate geometry as it describes the steepness and direction of a line, and it is foundational for analyzing linear relationships.

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Finding the Slope of a Line Example 3

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Finding the Slope of a Line Example 3 Video Summary

To find the slope of a line given by an equation, one effective method is to first determine the x-intercept and y-intercept. The x-intercept is found by setting y to zero and solving for x, while the y-intercept is found by setting x to zero and solving for y. For example, consider the equation 2x + 3y = 12. Setting y = 0 gives 2x = 12, so x = 6, which means the x-intercept is at the point (6, 0). Similarly, setting x = 0 gives 3y = 12, so y = 4, and the y-intercept is at (0, 4).

Once these intercepts are identified, they serve as two points on the line, allowing the calculation of the slope using the formula for slope between two points:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Here, using the points (6, 0) and (0, 4), the slope m is:

\[m = \frac{4 - 0}{0 - 6} = \frac{4}{-6} = -\frac{2}{3}\]

This negative slope indicates the line decreases as it moves from left to right. Understanding how to extract intercepts from an equation and then use them to find the slope reinforces the connection between algebraic expressions and their graphical representations. This approach is especially useful when no graph or explicit points are provided, enabling the determination of slope through fundamental algebraic manipulation and the concept of rise over run.

Problem

Find the slope of the following line.
$\frac{1}{2} x - \frac{2}{3} y = 4$

$\frac{3}{4}$

$- \frac{3}{4}$

$negative 6$

$negative 8$

Problem

Find the slope of the following line.
$0.5 x + 0.2 y = 3$

$\frac{2}{5}$

$\frac{6}{5}$

$\frac{5}{2}$

$- \frac{5}{2}$

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Types of Slope

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Types of Slope Video Summary

Understanding slopes is essential in mathematics, as it describes the steepness of a line or hill, represented as the ratio of the vertical change (rise) to the horizontal change (run). Mathematically, this is expressed as the formula:

\[\text{slope} = \frac{\Delta y}{\Delta x}\]

Slopes can be categorized into four types: positive, negative, zero, and undefined. Each type provides insight into the behavior of the line.

Positive slopes occur when a line rises from left to right. For example, if you select two points on a line where the vertical change (rise) is 2 and the horizontal change (run) is 1, the slope would be:

\[\text{slope} = \frac{2}{1} = 2\]

Negative slopes, on the other hand, indicate a line that falls from left to right. If the rise is -1 (indicating a downward movement) and the run is 1, the slope calculation would yield:

\[\text{slope} = \frac{-1}{1} = -1\]

For horizontal lines, the slope is always zero because there is no vertical change. For instance, if the y-values remain constant at 2 while the x-values change, the slope is:

\[\text{slope} = \frac{0}{\Delta x} = 0\]

This indicates a perfectly flat line, which is represented by the equation:

\[y = c\]

where $c$ is a constant value.

In contrast, vertical lines have an undefined slope because the horizontal change (run) is zero. For example, if both points lie on the line where $x = 2$, the slope calculation would be:

\[\text{slope} = \frac{\Delta y}{0}\]

Since division by zero is undefined, we conclude that the slope of vertical lines is also undefined. The equation for vertical lines is expressed as:

\[x = a\]

where $a$ is a constant value. Understanding these concepts is crucial for analyzing linear relationships in various mathematical contexts.

Problem

Graph a line with a slope of $0$ that passes through the point $open paren 3 comma negative 2 close paren$ .

Graphing coordinate with curve

Problem

Which of the following graphs below represents the equation $x equals 3$ ?

A graph of a line

Here’s what students ask on this topic:

The slope of a line, represented by $m$ , is calculated using the formula $\frac{y 2 - y 1}{x 2 - x 1}$ . This means you subtract the y-coordinate of the first point from the y-coordinate of the second point, then divide by the difference between the x-coordinates of the same points. This ratio, often called "rise over run," measures how steep the line is. It is important to keep the order consistent when subtracting to get the correct slope value.

Slopes describe the steepness and direction of a line. A positive slope means the line rises from left to right, indicating an increase in y as x increases. A negative slope means the line falls from left to right, showing a decrease in y as x increases. A zero slope corresponds to a horizontal line where y remains constant regardless of x; mathematically, this is $m = 0$ . An undefined slope occurs with vertical lines where x is constant and the change in x is zero, making the slope formula division by zero, which is undefined. Understanding these categories helps in graphing and analyzing linear equations.

To find the slope from a graph, first identify two points on the line with clear coordinates. Then calculate the "rise," which is the vertical change between the points, and the "run," which is the horizontal change. The slope is the ratio of rise over run, expressed as $\frac{Δy}{Δx}$ , where $Δy$ is the change in y-values and $Δx$ is the change in x-values. For example, if the line rises 3 units and runs 1 unit to the right, the slope is 3. This method visually shows how steep the line is.

A horizontal line has a slope of zero because there is no vertical change between any two points on the line. Mathematically, this means $m = 0$ . The equation of a horizontal line is written as $y = b$ , where $b$ is the constant y-value for all points on the line. This indicates that no matter what x-value you choose, y remains the same, resulting in a flat, level line.

The slope of a vertical line is undefined because the change in x-values ( $Δx$ ) is zero, and division by zero is undefined in mathematics. Since slope is calculated as $\frac{Δy}{Δx}$ , having $Δx = 0$ makes the slope undefined. The equation of a vertical line is written as $x = a$ , where $a$ is the constant x-value for all points on the line. This means the line runs straight up and down without any horizontal movement.

The order of points does not affect the value of the slope as long as you subtract the coordinates consistently. When using the formula $\frac{y 2 - y 1}{x 2 - x 1}$ , if you switch the points, you must subtract in the same order for both y and x values. For example, if you swap points, subtract the y-values in the new order and do the same for the x-values. This ensures the slope remains the same, reflecting the line's steepness correctly.