key term - $y - y_1 = m(x - x_1)$

Definition

$y - y_1 = m(x - x_1)$ is an equation that represents the slope-intercept form of a linear equation in two variables, $x$ and $y$. It describes the relationship between the coordinates of a point $(x_1, y_1)$ on the line and the slope $m$ of the line. This equation is fundamental in graphing linear equations in the Cartesian coordinate system.

5 Must Know Facts For Your Next Test

1. The slope-intercept form, $y - y_1 = m(x - x_1)$, allows you to easily identify the slope and a point on the line to graph the linear equation.
2. The slope, $m$, represents the rate of change between the $x$- and $y$-variables, indicating how much $y$ changes for a unit change in $x$.
3. The point $(x_1, y_1)$ represents a known point on the line, which can be used to determine the $y$-intercept of the linear equation.
4. The slope-intercept form can be rearranged to the standard form of a linear equation, $Ax + By = C$, by solving for $y$ in terms of $x$.
5. Understanding the slope-intercept form is crucial for graphing linear equations and interpreting the relationships between variables in the Cartesian coordinate system.

Review Questions

• Explain how the slope-intercept form, $y - y_1 = m(x - x_1)$, can be used to graph a linear equation.
  • The slope-intercept form, $y - y_1 = m(x - x_1)$, provides the necessary information to graph a linear equation in the Cartesian coordinate system. The slope, $m$, determines the steepness and direction of the line, while the point $(x_1, y_1)$ represents a known point on the line. By plotting this point and using the slope to determine the direction and rate of change, you can accurately graph the linear equation.
• Describe how the slope-intercept form, $y - y_1 = m(x - x_1)$, can be used to determine the standard form of a linear equation, $Ax + By = C$.
  • To convert the slope-intercept form, $y - y_1 = m(x - x_1)$, to the standard form of a linear equation, $Ax + By = C$, you can rearrange the terms. First, isolate the $y$ term on the left-hand side: $y = mx - mx_1 + y_1$. Then, collect the coefficients of $x$ and $y$ to obtain the standard form: $y = mx - mx_1 + y_1$ can be written as $-m x + y = mx_1 - y_1$, where $A = -m$, $B = 1$, and $C = mx_1 - y_1$.
• Analyze how the slope-intercept form, $y - y_1 = m(x - x_1)$, can be used to interpret the relationship between the variables $x$ and $y$ in a linear equation.
  • The slope-intercept form, $y - y_1 = m(x - x_1)$, provides valuable insights into the relationship between the variables $x$ and $y$ in a linear equation. The slope, $m$, represents the rate of change, indicating how much the $y$-value changes for a unit change in the $x$-value. This information can be used to understand the proportional relationship between the variables and make predictions about how changes in one variable will affect the other. Additionally, the point $(x_1, y_1)$ represents a known coordinate on the line, which can be used to determine the $y$-intercept and further analyze the behavior of the linear equation.

Related terms

Slope:

The slope, $m$, represents the rate of change or steepness of a line. It indicates how much the $y$-value changes for a unit change in the $x$-value.

Point-Slope Form:

The point-slope form of a linear equation is another way to express the relationship between a point on the line and the slope, written as $y - y_1 = m(x - x_1)$.

Cartesian Coordinate System: The Cartesian coordinate system is a two-dimensional plane used to graph linear equations, where the $x$-axis represents the horizontal dimension and the $y$-axis represents the vertical dimension.