Ax + By = C

Definition

The equation Ax + By = C represents a linear equation in two variables, where A, B, and C are constants, and x and y are the variables. This equation defines a straight line in the coordinate plane and is a fundamental concept in the study of linear functions and their graphical representations.

5 Must Know Facts For Your Next Test

1. The constants A, B, and C in the equation Ax + By = C define the unique characteristics of the line, such as its slope and y-intercept.
2. The slope of the line represented by Ax + By = C can be calculated as -A/B, which is the negative ratio of the coefficients of x and y.
3. The y-intercept of the line can be found by solving the equation Ax + By = C for y when x = 0, yielding the y-intercept as C/B.
4. The slope-intercept form of the equation Ax + By = C is y = (-A/B)x + (C/B), which explicitly shows the slope and y-intercept of the line.
5. The equation Ax + By = C can be used to represent a wide variety of linear relationships, including those that model real-world phenomena, such as the relationship between distance, time, and speed.

Review Questions

• Explain how the coefficients A, B, and C in the equation Ax + By = C determine the characteristics of the line.
  • The coefficients A, B, and C in the equation Ax + By = C define the unique characteristics of the line. The ratio of -A/B determines the slope of the line, while the ratio of C/B gives the y-intercept. These coefficients, along with the variables x and y, collectively describe the linear relationship represented by the equation and its graphical representation in the coordinate plane.
• Describe the process of converting the general equation Ax + By = C into the slope-intercept form y = mx + b.
  • To convert the general equation Ax + By = C into the slope-intercept form y = mx + b, you need to solve the equation for y. This involves dividing both sides of the equation by the coefficient of y, B, to isolate y on one side. The resulting equation will be in the form y = (-A/B)x + (C/B), where -A/B represents the slope (m) and C/B represents the y-intercept (b). This conversion allows you to more easily identify the key characteristics of the line, such as its slope and y-intercept, from the general equation.
• Explain how the equation Ax + By = C can be used to model real-world linear relationships, and provide an example of such a relationship.
  • The general equation Ax + By = C can be used to model a wide variety of linear relationships in the real world. For example, the equation could represent the relationship between distance, time, and speed, where A represents the speed, B represents the time, and C represents the distance traveled. In this case, the equation would take the form d = vt, where d is the distance, v is the speed, and t is the time. By rearranging the equation into the slope-intercept form, you can easily determine the speed (slope) and the initial position (y-intercept) of an object in motion. This demonstrates the versatility of the Ax + By = C equation in modeling and analyzing linear relationships encountered in various fields, such as physics, economics, and engineering.

Related terms

Slope:

The slope of a line is a measure of the steepness of the line, and it is typically represented by the letter 'm'. The slope can be calculated as the change in y-value divided by the change in x-value between any two points on the line.

Y-intercept: The y-intercept of a line is the point where the line crosses the y-axis, and it is typically represented by the letter 'b'. The y-intercept can be found by substituting x = 0 into the linear equation Ax + By = C.

Slope-Intercept Form:

The slope-intercept form of a linear equation is y = mx + b, where m represents the slope of the line and b represents the y-intercept. This form is derived from the general equation Ax + By = C by solving for y in terms of x.