Question

what is the slope of this line? simplify your answer and write it as a proper fraction, improper fraction, or integer.

Explanation:

Step1: Recuerda la fórmula de la pendiente

La fórmula para la pendiente \( m \) entre dos puntos \( (x_1, y_1) \) y \( (x_2, y_2) \) es \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Identifica los puntos

Tenemos los puntos \( (-1, -2) \) y \( (1, 4) \). Asignamos \( (x_1, y_1) = (-1, -2) \) y \( (x_2, y_2) = (1, 4) \).

Step3: Sustituye en la fórmula

Sustituimos los valores en la fórmula: \( m = \frac{4 - (-2)}{1 - (-1)} \).

Step4: Simplifica el numerador y el denominador

El numerador: \( 4 - (-2) = 4 + 2 = 6 \). El denominador: \( 1 - (-1) = 1 + 1 = 2 \).

Step5: Calcula la pendiente

Entonces, \( m = \frac{6}{2} = 3 \)? Espera, no, espera. Espera, me equivoqué. Espera, \( 4 - (-2) = 6 \), \( 1 - (-1) = 2 \), \( 6/2 = 3 \)? Wait, no, wait the points: Wait, the graph: Wait, maybe I mixed up x and y? Wait, the axes: Wait, the x-axis is vertical? Wait, no, the standard coordinate system: Wait, the graph has x-axis (vertical) and y-axis (horizontal)? Wait, no, that's unusual. Wait, the labels: the x-axis is the vertical one (downward is positive x?), and y-axis is horizontal (right is positive y?). Wait, maybe the coordinates are (x, y) where x is vertical and y is horizontal. Wait, let's check the points: (-1, -2): x=-1 (vertical), y=-2 (horizontal). (1, 4): x=1, y=4. Then the slope formula: if x is vertical and y is horizontal, then the slope would be \( \frac{y_2 - y_1}{x_2 - x_1} \)? Wait, no, maybe the axes are swapped. Wait, maybe it's a typo, but let's proceed with the given points. Wait, original points: (-1, -2) and (1, 4). So \( x_1 = -1 \), \( y_1 = -2 \); \( x_2 = 1 \), \( y_2 = 4 \). Then slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-2)}{1 - (-1)} = \frac{6}{2} = 3 \). Wait, but that seems steep. Wait, maybe I got the points wrong. Wait, looking at the graph: the blue line passes through (-1, -2) and (1, 4)? Wait, no, maybe the points are ( -2, -1) and (4, 1)? Wait, no, the labels: (-1, -2) is marked, and (1, 4)? Wait, maybe the axes are swapped. Wait, maybe the horizontal axis is y and vertical is x. So in standard terms, if horizontal is y and vertical is x, then the slope would be \( \frac{\Delta x}{\Delta y} \)? No, the slope is rise over run, where rise is change in y (vertical) and run is change in x (horizontal). Wait, maybe the graph has x-axis vertical (so x is vertical) and y-axis horizontal (y is horizontal). So in that case, the slope (which is rise over run, where rise is change in x (vertical) and run is change in y (horizontal)). Wait, that's non-standard, but let's check. So if x is vertical (up is positive x) and y is horizontal (right is positive y). Then the two points: (-1, -2): x=-1 (down 1), y=-2 (left 2). (1, 4): x=1 (up 1), y=4 (right 4). Then the slope (change in x over change in y) would be \( \frac{1 - (-1)}{4 - (-2)} = \frac{2}{6} = \frac{1}{3} \). Ah! That's the mistake. I mixed up x and y axes. Because in the graph, the x-axis is vertical (the vertical line with arrow down) and y-axis is horizontal (the horizontal line with arrow right). So the coordinates are (x, y) where x is vertical (so x increases downward) and y is horizontal (y increases to the right). So the two points: (-1, -2): x=-1 (so 1 unit up from origin), y=-2 (2 units left from origin). (1, 4): x=1 (1 unit down from origin), y=4 (4 units right from origin). So the change in x (Δx) is \( 1 - (-1) = 2 \) (down 2 units), and change in y (Δy) is \( 4 - (-2) = 6 \) (right 6 units). Then the slope (which is Δx / Δy, because x is vertical) would be \( \frac{2}{6} = \frac{1}{3} \)? Wait, no, slope is usually defined as (change in y)/(change in x) where y is vertical and x is horizontal. But here the axes are swapped. So maybe the problem has a non-standard coordinate system. Wait, the original problem: "What is the slope of this line?" with the graph. Let's re-express the points correctly. Let's assume that the horizontal axis is y (right is positive y) and vertical axis is x (down is positive x). So the two points: First point: (-1, -2): x=-1 (up 1), y=-2 (left 2). Second point: (1, 4): x=1 (down 1), y=4 (right 4). So the change in y (Δy) is 4 - (-2) = 6 (right 6), change in x (Δx) is 1 - (-1) = 2 (down 2). Then slope (rise over run) where rise is Δx (vertical change) and run is Δy (horizontal change) would be Δx / Δy = 2/6 = 1/3. Wait, but that seems low. Wait, maybe I got the points wrong. Wait, looking at the graph: the blue line passes through (-2, -1) and (4, 1)? Wait, no, the labels are (-1, -2) and (1, 4). Wait, maybe the axes are standard, but the graph is drawn with x vertical. So in standard terms, if we consider the horizontal axis as x and vertical as y, but the graph is rotated. Wait, this is confusing. Wait, let's use the formula correctly. Let's take the two points as (x1, y1) = (-1, -2) and (x2, y2) = (1, 4), regardless of axes. Then slope m = (y2 - y1)/(x2 - x1) = (4 - (-2))/(1 - (-1)) = 6/2 = 3. But that would be a steep line. Wait, but the graph: the line goes from (-1, -2) to (1, 4). Let's plot these points: (-1, -2) is left 1, down 2; (1, 4) is right 1, up 4. So the line goes from lower left to upper right, with a rise of 6 (from y=-2 to y=4) and run of 2 (from x=-1 to x=1). So slope is 6/2 = 3. Yes, that makes sense. So my initial calculation was correct. So the slope is 3? Wait, but let's check again. 4 - (-2) = 6, 1 - (-1) = 2, 6 divided by 2 is 3. Yes.

Answer:

3