Question

A point on a line and its slope are given. Find the point-slope form of the equation of the line. $$ P=(-1,3) ; m=0 $$

Short answer

y = 3

Step-by-step solution

Identify given values

Identify and list the given point and slope. Here, the point (P) is P(-1, 3) and the slope (m) is 0.

Substitute into point-slope form equation

The point-slope form of the equation of a line is given by y - y_1 = m(x - x_1).Substitute the given point and slope into this equation: y - 3 = 0(x + 1).

Simplify the equation

Simplify the equation obtained from the substitution: y - 3 = 0. Simplifying further results in: y = 3.

Key concepts

line equation

In mathematics, a line equation is an essential tool to describe the relationship between the x and y coordinates on a Cartesian plane. A line equation can take multiple forms, such as slope-intercept form, point-slope form, and standard form. Each form offers unique ways to understand and work with the properties of a line.

Slope-intercept form is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. It is useful for quickly identifying the slope and y-intercept of a line.

The point-slope form, which is the focus of our exercise, is expressed as \( y - y_1 = m(x - x_1) \). This form is particularly helpful when we know a point on the line and the slope but are not as concerned with the intercepts.
Finally, the standard form of a line equation is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. This form is useful in more advanced algebraic manipulation and solving systems of equations.

slope

The slope of a line measures its steepness and direction, determining how much the line rises or falls as it moves along the x-axis. The slope is represented by the letter \( m \) and can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
This equation takes two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line and determines the ratio of the vertical change to the horizontal change between these points.

A positive slope means the line rises as it moves from left to right. Conversely, a negative slope indicates the line falls.
If the slope is zero, the line is horizontal and does not rise or fall. This is the case in our example with the given slope \( m=0 \).

By understanding the slope, it becomes easier to predict and analyze the behavior of lines on a graph.

simplifying equations

Simplifying equations is a critical step in solving algebraic problems, as it makes the equations easier to work with.

In our exercise, after substituting the given point and slope into the point-slope form, we obtained the equation \( y - 3 = 0(x + 1) \). This equation was then simplified by multiplying and combining like terms.

The first step involved recognizing that any term multiplied by zero is zero, so \( 0(x + 1) \) simplifies directly to zero. Thus, the equation reduces to \( y - 3 = 0 \).
The final step is to isolate the variable \( y \) by performing algebraic operations.
Adding 3 to both sides, we get \( y = 3 \).

These simplified steps help focus on the core information that defines the equation of the line and makes it easier to interpret and use.

point-slope formula

The point-slope formula is a crucial concept in algebra that allows you to write the equation of a line when you know a point on the line and its slope. The formula is written as:
\( y - y_1 = m(x - x_1) \).
The variables \( (x_1, y_1) \) represent the coordinates of the given point, and \( m \) stands for the slope of the line.

In our problem, we are given the point \( P(-1, 3) \) and the slope \( m=0 \). Substituting these values into the point-slope formula, we get:
¨ \( y - 3 = 0(x + 1) \).

Since \( 0 \) times any value is zero, this simplifies to \( y - 3 = 0 \).
By performing basic algebraic operations, we find that \( y = 3 \).

This resulting equation tells us that the line is horizontal and passes through the point (0,3).

Thus, the point-slope formula is a powerful tool for quickly deriving the equation of a line when given critical initial information.