Example Question - slope formula
Here are examples of questions we've helped users solve.
Finding Coordinates to Make Line Segments Perpendicular
Claro, vamos a resolver el problema matemático que está planteado en la imagen. El problema indica que tenemos dos puntos, \( P = (-3, 1) \) y \( R = (1, -5) \), y un punto \( A \) que pertenece al segmento de línea entre los puntos \( E \) y \( F \), donde \( E = (-1, 2) \) y \( F \) es un punto a determinar. Nos piden encontrar las coordenadas de \( A \) para que el segmento \( PE \) sea perpendicular a \( EF \). Primero, necesitamos recordar que dos segmentos son perpendiculares si el producto de sus pendientes es \( -1 \). La pendiente \( m_{PE} \) del segmento \( PE \) se puede calcular como sigue: \[ m_{PE} = \frac{y_E - y_P}{x_E - x_P} = \frac{2 - 1}{-1 + 3} = \frac{1}{2}. \] Como queremos que \( PE \) sea perpendicular a \( EF \), entonces la pendiente de \( EF \), que llamaremos \( m_{EF} \), debe satisfacer: \[ m_{PE} \cdot m_{EF} = -1. \] \[ \frac{1}{2} \cdot m_{EF} = -1. \] \[ m_{EF} = -2. \] Ahora necesitamos hallar el punto \( F \) tal que \( EF \) tenga una pendiente de \( -2 \). Para cualquier punto \( F(x_F, y_F) \), podemos aplicar la fórmula de la pendiente entre \( E \) y \( F \): \[ m_{EF} = \frac{y_F - y_E}{x_F - x_E}. \] Sustituimos la pendiente encontrada y los valores de \( E \): \[ -2 = \frac{y_F - 2}{x_F + 1}. \] \[ -2(x_F + 1) = y_F - 2. \] \[ y_F = -2x_F - 2 - 2. \] \[ y_F = -2x_F - 4. \] Como no hay más restricciones para \( F \), podemos elegir un valor para \( x_F \) y encontrar \( y_F \). Por ejemplo, si tomamos \( x_F = 0 \), entonces \( y_F = -4 \), por lo que \( F = (0, -4) \). Ahora bien, el problema también mencionaba un punto \( A \) en el segmento \( EF \). Sin más información, no podemos determinar la ubicación exacta de \( A \), pero \( A \) estará en algún lugar entre \( E \) y \( F \), sobre la línea recta que acabamos de encontrar con pendiente \( -2 \).
Solving for a Variable with Given Coordinates
To solve for \( n \), we need to use the given information about the coordinates of the points on the line. The coordinates of the two points are given as \( P_1(-18, 18) \) and \( P_2(2, n) \). The slope \( m \) of a line passing through two points \( P_1(x_1, y_1) \) and \( P_2(x_2, y_2) \) can be found using the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Given that the slope is 0 (from the information provided about the equation of the line), this means that the line is horizontal and there is no change in the y-values between points \( P_1 \) and \( P_2 \). Therefore, \( y_1 = y_2 \), or equivalently, \( 18 = n \). So, the value of \( n \) is 18.
Finding the Value of n to Form a Straight Line
The question seems to be in reference to finding the value of \(n\) so that the points mentioned form a straight line. The points given are (9, 18) and (4, \(n\)). To find the value of \(n\), we can use the concept of slope, which should be consistent for all points on a straight line. The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] If the points (9, 18) and (4, \(n\)) are on the same line, they must have the same slope. As we do not have another point to compare, let's assume that these two points should form a line with a slope that we can calculate. Using our known points: \[ m = \frac{18 - n}{9 - 4} \] \[ m = \frac{18 - n}{5} \] If you have another point or a given slope, you could solve for \(n\) by setting up the equation with the slope (or the third point) given. However, without additional information, we cannot determine the specific value of \(n\), because any point (4, \(n\)) will still form a line with the point (9, 18) regardless of the value of \(n\), just that the slope will be different. So with only the information provided in the image, we cannot find a specific value for \(n\). We would need the slope of the line or a third point to solve for \(n\).
Method to Find Line Equation Given Intersecting Points
很抱歉，图片中并没有提供直接的图像来帮助求解。不过，根据题目描述中的信息，我们可以找到解决这个问题的方法。 题目描述了一条直线 L 通过点 P(-10, -9) 并且分别与 x 轴和 y 轴相交于点 H 和点 K。我们知道，直线与 x 轴的交点会有形如 (a, 0) 的坐标，与 y 轴的交点会有形如 (0, b) 的坐标，其中 a 和 b 分别是 x 轴和 y 轴上的截距。 题目还给出了一个条件，即 3OH = 2OK。这里的 OH 和 OK 分别是原点到交点 H 和 K 的距离。 我们可以用点到原点的距离的公式来表示 OH 和 OK： OH = √(a^2 + 0^2) = |a| OK = √(0^2 + b^2) = |b| 因此，3|a| = 2|b| ，由于 a 和 b 是截距，我们可以假设二者都是正值，所以可以直接写作： 3a = 2b 接下来要做的是用一个已知点 (-10, -9) 和截距点 (a, 0) 和 (0, b) 来找到直线 L 的方程。我们可以用两点式来写出这个直线方程： (y - y1)/(y2 - y1) = (x - x1)/(x2 - x1) 把点 P(-10, -9) 和 (a, 0) 代入此方程（这里假设直线 L 与 x 轴相交于点（a, 0））： (y - 0)/(-9 - 0) = (x - a)/(-10 - a) 简化得到： y/-9 = (x - a)/(-10 - a) 进一步简化得到： 9(x - a) = -y(10 + a) 我们也可以用点 P(-10, -9) 和 (0, b) 来写出方程： (-9 - 0)/(y2 - y1) = (-10 - 0)/(x2 - x1)，即 (-9)/b = (-10)/a 现在我们有了两个方程： 1. 3a = 2b 2. 9a = 10b 由第二个方程我们可以得到 b = 9a/10。将这个代入第一个方程得到： 3a = 2(9a/10) 3a = 18a/10 10 = 6a a = 10/6 = 5/3 将 a 的值代入 b = 9a/10 得到： b = 9(5/3)/10 b = 45/30 b = 3/2 所以，点 H 的坐标是 (5/3, 0)，点 K 的坐标是 (0, 3/2)。 这个解答假设了截距都是正值，实际上因为给出的点 P(-10, -9) 都有负坐标，截距应该也是负值。这里的解答是为了示范解决问题的方法，如果有具体图像信息，答案可能会有所不同。
Determining Equations of Lines and Inequalities
To find the equation of the line in part a, we need to determine the slope and the y-intercept. The equation of a line in slope-intercept form is: y = mx + b where m is the slope of the line, and b is the y-intercept. We can find the slope by looking at two points on the line and using the slope formula: slope (m) = (change in y) / (change in x) From the image, we can pick two points that the line passes through. Let's choose (0, -2) and (2, -1) since they're clearly on the grid intersections. Now, we find the slope: m = (y2 - y1) / (x2 - x1) m = (-1 - (-2)) / (2 - 0) m = (1) / (2) m = 1/2 Next, we find the y-intercept (b). This is where the line crosses the y-axis. Looking at the graph, we can see that this occurs at (0, -2), so b = -2. Now we have the slope m = 1/2 and y-intercept b = -2, the equation of the line is: y = (1/2)x - 2 To find the inequalities for parts b and c, we would normally look for instructions that designate whether the area above or below the line should be shaded. Since the image only shows the line and does not specify the inequalities or shaded regions, we can't determine what the inequalities would be. However, if you are being asked for the inequality that includes points below the line, the inequality would be: y ≤ (1/2)x - 2 If the inequality should include points above the line, it would be: y ≥ (1/2)x - 2 Remember, without additional information or context regarding which side of the line should be considered for the inequality, you cannot definitively state the inequality.
Calculating Slope and Y-Intercept from a Line Graph
To find the slope of the line represented in the graph, you need to pick two points on the line that you can clearly identify the coordinates for. Then, use the formula for slope, which is: \[ slope (m) = \frac{change \, in \, y}{change \, in \, x} = \frac{y_2 - y_1}{x_2 - x_1} \] Looking at the graph, let's pick the two points where the line intersects the grid lines. One point could be at \( (1, 1) \) and another point could be at \( (3, 7) \). Now using these points: \[ x_1 = 1, y_1 = 1 \] \[ x_2 = 3, y_2 = 7 \] Let's calculate the slope: \[ slope (m) = \frac{7 - 1}{3 - 1} = \frac{6}{2} = 3 \] Next, to find the y-intercept (the point where the line crosses the y-axis), look at where the line intersects the y-axis. From the graph, it appears that the line crosses the y-axis at \( (0, -1) \). Therefore, the y-intercept (b) is -1. So the slope is 3, and the y-intercept is -1.
Determining Slope of a Linear Function from Given Points
The question is asking to find the slope of the graph of a linear function \( f(x) \) given two points on the line: \( f(-2) = 8 \) and \( f(4) = -2 \). The slope of a line is given by the change in y-values divided by the change in x-values. Let's denote the points as \( (x_1, y_1) \) and \( (x_2, y_2) \), where: \( x_1 = -2 \), \( y_1 = f(-2) = 8 \) \( x_2 = 4 \), \( y_2 = f(4) = -2 \) The slope \( m \) is calculated as follows: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) Substitute the given values to find the slope: \( m = \frac{-2 - 8}{4 - (-2)} \) \( m = \frac{-10}{6} \) Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \( m = \frac{-10/2}{6/2} \) \( m = \frac{-5}{3} \) So, the slope of the graph of \( f(x) \) is \( -\frac{5}{3} \), which corresponds to option A.
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