Question

A point on a line and its slope are given. Find the point-slope form of the equation of the line. $$ P=(2,1) ; m=4 $$

Short answer

The point-slope form of the equation is y - 1 = 4(x - 2)

Step-by-step solution

- Understand the point-slope form equation

The point-slope form of the equation of a line is given by y - y_1 = m(x - x_1)where (x_1, y_1) is a point on the line and m is the slope of the line.

- Substitute the given point and slope

Use the point P = (2, 1) and the slope m = 4. Substitute these values into the formula: y - 1 = 4(x - 2)

- Simplify the expression if necessary

In this case, the equation is already in the required point-slope form, so no further simplification is necessary. The final equation is: y - 1 = 4(x - 2)

Key concepts

Equation of a Line

In mathematics, an equation of a line is a formula that represents all the points on a given straight line on a graph. A line can be described using various forms, including slope-intercept form, point-slope form, and standard form. The type of equation you use depends on the information you have. In this exercise, we focus on the point-slope form because we know a point on the line and the line's slope.
The point-slope form is particularly useful for quickly writing the equation of the line when you know a specific point the line passes through and its slope.

Slope

The slope of a line measures its steepness and direction. In mathematical terms, the slope 'm' is defined as the ratio of the vertical change (the rise) to the horizontal change (the run) between two points on the line.
It's calculated as: \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\).
A positive slope means the line rises as it moves from left to right, while a negative slope means the line falls. A zero slope indicates a horizontal line, and an undefined slope corresponds to a vertical line. In the given exercise, the slope is 4, meaning the line rises steeply.

Point on a Line

A point on a line is a specific location defined by its coordinates (x,y) that lies on the line itself. In this exercise, the given point is P = (2,1).
Points are crucial in defining the line because, along with the slope, they allow us to find the equation of that specific line. In point-slope form, we use this point to accurately place the line in the graph. This is seen in the equation y - y_1 = m(x - x_1), where (x_1, y_1) represents the given point.
Substituting the values helps us write the precise equation describing the line.

Linear Equations

Linear equations are equations that form straight lines when graphed. They have the highest exponent of 1 in their variables. A typical form of a linear equation in two variables is: \(Ax + By + C = 0\).
Linear equations help explain relationships where changes in one variable result in proportionate changes in another. They can be used to model real-world scenarios, such as predicting expenses, calculating distances, or understanding speed. In the point-slope form as used here, linear equations make it simple to determine the line's equation when you know one point and the line's slope. By substituting the given point and slope into the point-slope equation, we derived \( y - 1 = 4(x - 2)\).