key term - Point-Slope Form

Definition

The point-slope form is a way to represent a linear equation by specifying a point on the line and the slope of the line. It allows for the equation of a line to be expressed in a compact and intuitive manner, making it a useful tool for analyzing and graphing linear relationships.

5 Must Know Facts For Your Next Test

1. The point-slope form of a linear equation is typically written as $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the slope of the line.
2. The point-slope form allows you to easily determine the equation of a line given a point on the line and the slope of the line.
3. Rearranging the point-slope form can yield the slope-intercept form of a linear equation, $y = mx + b$, where $b$ is the y-intercept.
4. The point-slope form is particularly useful when the slope of a line is known, but the y-intercept is not.
5. Point-slope form can be used to graph a linear equation by plotting the given point and then using the slope to determine the direction and steepness of the line.

Review Questions

• Explain how the point-slope form of a linear equation is derived from the general linear equation $y = mx + b$.
  • The point-slope form, $y - y_1 = m(x - x_1)$, is derived from the general linear equation $y = mx + b$ by rearranging the terms. If we know a point $(x_1, y_1)$ on the line and the slope $m$, we can substitute these values into the general equation to solve for the y-intercept $b$. Rearranging the equation to isolate $y$ on one side and $x$ on the other results in the point-slope form, which expresses the linear equation in terms of the slope and a known point on the line.
• Describe how the point-slope form can be used to graph a linear equation.
  • The point-slope form, $y - y_1 = m(x - x_1)$, can be used to graph a linear equation by first plotting the known point $(x_1, y_1)$ on the coordinate plane. Then, using the slope $m$, you can determine the direction and steepness of the line. By starting at the point $(x_1, y_1)$ and moving along the line in the direction indicated by the slope, you can plot additional points to construct the complete linear graph.
• Analyze how the point-slope form can be used to solve real-world problems involving linear relationships.
  • The point-slope form is particularly useful in real-world applications where the slope of a linear relationship is known, but the y-intercept is not. For example, in economics, the point-slope form can be used to model the supply or demand curve for a product, where the slope represents the rate of change in price with respect to quantity. In physics, the point-slope form can be used to describe the motion of an object, where the slope represents the velocity or acceleration. By using the point-slope form, you can determine the equation of the line and make predictions or analyze the linear relationship without needing to know the y-intercept.

Related terms

Slope:

The slope of a line is a measure of its steepness, representing the change in the y-coordinate for a unit change in the x-coordinate.

Intercept:

The y-intercept of a line is the point where the line crosses the y-axis, representing the value of y when x is equal to 0.

Linear Equation:

A linear equation is an equation that represents a straight line, where the variables are raised to the power of 1 and are connected by addition or subtraction.