Example Question - slope of a line
Here are examples of questions we've helped users solve.
Finding Equation of Perpendicular Line through Point
To find the equation of the line h, which is perpendicular to line g and passes through the point (3, -9), we need to start by identifying the slope of line g. The equation for line g is written in slope-intercept form as: \[ y = \frac{3}{10}x - 8 \] The slope (m) of line g is the coefficient of x, which is \(\frac{3}{10}\). For a line to be perpendicular to another line, its slope must be the negative reciprocal of the original line's slope. The negative reciprocal of \(\frac{3}{10}\) is \(-\frac{10}{3}\) (because when you flip the fraction and change the sign, you get the negative reciprocal). Now that we have the slope of the perpendicular line, we can use the point-slope form of the equation of a line to find line h's equation. Point-slope form is: \[ y - y_1 = m(x - x_1) \] where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Using the slope of line h, which is \(-\frac{10}{3}\), and the given point (3, -9), we plug these into the point-slope form: \[ y - (-9) = -\frac{10}{3}(x - 3) \] Simplify the equation: \[ y + 9 = -\frac{10}{3}x + \frac{10}{3} \cdot 3 \] \[ y + 9 = -\frac{10}{3}x + 10 \] We want to get it into slope-intercept form (y = mx + b), so we'll isolate y: \[ y = -\frac{10}{3}x + 10 - 9 \] \[ y = -\frac{10}{3}x + 1 \] The equation of line h in slope-intercept form, with numbers as simplified proper fractions, improper fractions, or integers, is: \[ y = -\frac{10}{3}x + 1 \]
Finding Parallel Lines in Coordinate Geometry
Câu hỏi yêu cầu ta tìm giá trị của "a" để đường thẳng d': y = ax - 19 song song với đường thẳng d: y = 3x + 4. Để hai đường thẳng song song với nhau, hệ số góc (hay còn gọi là hệ số của x) của chúng phải bằng nhau. Hệ số góc của đường thẳng d là 3. Như vậy, để d' song song với d, hệ số góc của d' cũng phải là 3. Điều này có nghĩa là a phải bằng 3. Vậy đáp án là C. a = 3.
Derivative Calculation with a Function and Point on a Graph
The image includes a graph displaying a straight line representing a function y = f(x), along with a pair of points labeled A(2, 1). Additionally, there's a mathematical question that states: Assuming the function g(x + 1) = (x^2 + 1) - f(x), what is the value of g'(3)? First, let's determine the slope of the line representing y = f(x) since we have the point A(2, 1) on the line. Since the line passes through the point A(2, 1) and the origin (0, 0), we can use these two points to calculate the slope (m): m = (y2 - y1) / (x2 - x1) m = (1 - 0) / (2 - 0) m = 1 / 2 Thus, the slope of the line is 1/2, and knowing that a line equation with slope m passing through the origin (0,0) has the form y = mx, the equation for f(x) is: f(x) = (1/2)x Now, let's put f(x) into the equation for g(x + 1): g(x + 1) = (x^2 + 1) - (1/2)x To find g'(x), the derivative of g(x), we must first write g(x) in terms of x: g(x) = ((x - 1)^2 + 1) - (1/2)(x - 1) g(x) = (x^2 - 2x + 1 + 1) - (1/2)x + (1/2) g(x) = x^2 - (5/2)x + 2 The next step is to differentiate g(x) with respect to x to find g'(x): g'(x) = 2x - 5/2 Finally, to find g'(3), substitute 3 into the derivative: g'(3) = 2(3) - 5/2 g'(3) = 6 - 5/2 g'(3) = (12/2) - (5/2) g'(3) = (12 - 5)/2 g'(3) = 7/2 Therefore, g'(3) is 7/2.
Calculating Gradient (Slope) of a Line
To find the gradient (slope) of the line joining the points A(2, 3) and B(5, 9), you can use the formula for the gradient (m) which is the change in the y-coordinates divided by the change in the x-coordinates: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Given the points A(2, 3) and B(5, 9), you can label them as follows: \( (x_1, y_1) = (2, 3) \) \( (x_2, y_2) = (5, 9) \) Now plug these values into the formula: \[ m = \frac{9 - 3}{5 - 2} \] \[ m = \frac{6}{3} \] \[ m = 2 \] So the gradient of the line is 2.
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