Find The Slope And \[$ Y \$\]-intercept Of The Following Linear Equation. Express The \[$ Y \$\]-intercept As A Coordinate Pair.$\[ Y = \frac{2}{3} X - 8 \\]Answer:Slope: \[$\frac{2}{3}\$\]\[$ Y \$\]-intercept:
Introduction
In mathematics, a linear equation is a type of equation in which the highest power of the variable(s) is 1. It is a fundamental concept in algebra and is used to model various real-world situations. One of the key aspects of a linear equation is its slope and y-intercept. In this article, we will learn how to find the slope and y-intercept of a linear equation and express the y-intercept as a coordinate pair.
What is a Linear Equation?
A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form:
y = mx + b
where:
- y is the dependent variable
- x is the independent variable
- m is the slope of the line
- b is the y-intercept
Finding the Slope
The slope of a linear equation is a measure of how steep the line is. It is calculated as the ratio of the change in the dependent variable (y) to the change in the independent variable (x). In other words, it is the rate at which the line rises or falls as we move along the x-axis.
To find the slope of a linear equation, we can use the following formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Finding the y-intercept
The y-intercept of a linear equation is the point at which the line intersects the y-axis. It is the value of y when x is equal to 0. In other words, it is the point on the line where the x-coordinate is 0.
To find the y-intercept of a linear equation, we can use the following formula:
b = y - mx
where m is the slope of the line and x is the x-coordinate of the point at which the line intersects the y-axis.
Example: Finding the Slope and y-intercept of a Linear Equation
Let's consider the following linear equation:
y = (2/3)x - 8
To find the slope and y-intercept of this equation, we can use the formulas mentioned above.
Finding the Slope
To find the slope of this equation, we can use the formula:
m = (y2 - y1) / (x2 - x1)
Since the equation is in the form y = mx + b, we can see that the slope is already given as 2/3.
Finding the y-intercept
To find the y-intercept of this equation, we can use the formula:
b = y - mx
Since the equation is in the form y = mx + b, we can see that the y-intercept is already given as -8.
Expressing the y-intercept as a Coordinate Pair
The y-intercept is the point at which the line intersects the y-axis. It is the value of y when x is equal to 0. In other words, it is the point on the line where the x-coordinate is 0.
To express the y-intercept as a coordinate pair, we can write it as:
(0, -8)
This means that the y-intercept is the point (0, -8) on the coordinate plane.
Conclusion
In this article, we learned how to find the slope and y-intercept of a linear equation and express the y-intercept as a coordinate pair. We used the formulas for finding the slope and y-intercept and applied them to a given linear equation. We also expressed the y-intercept as a coordinate pair. This knowledge is essential for understanding linear equations and is used in various real-world applications.
Key Takeaways
- A linear equation is an equation in which the highest power of the variable(s) is 1.
- The slope of a linear equation is a measure of how steep the line is.
- The y-intercept of a linear equation is the point at which the line intersects the y-axis.
- To find the slope and y-intercept of a linear equation, we can use the formulas m = (y2 - y1) / (x2 - x1) and b = y - mx.
- The y-intercept can be expressed as a coordinate pair (0, b).
Frequently Asked Questions
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1.
Q: What is the slope of a linear equation?
A: The slope of a linear equation is a measure of how steep the line is.
Q: What is the y-intercept of a linear equation?
A: The y-intercept of a linear equation is the point at which the line intersects the y-axis.
Q: How do I find the slope and y-intercept of a linear equation?
A: To find the slope and y-intercept of a linear equation, you can use the formulas m = (y2 - y1) / (x2 - x1) and b = y - mx.
Q: How do I express the y-intercept as a coordinate pair?
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form:
y = mx + b
where:
- y is the dependent variable
- x is the independent variable
- m is the slope of the line
- b is the y-intercept
Q: What is the slope of a linear equation?
A: The slope of a linear equation is a measure of how steep the line is. It is calculated as the ratio of the change in the dependent variable (y) to the change in the independent variable (x). In other words, it is the rate at which the line rises or falls as we move along the x-axis.
Q: How do I find the slope of a linear equation?
A: To find the slope of a linear equation, you can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
Q: What is the y-intercept of a linear equation?
A: The y-intercept of a linear equation is the point at which the line intersects the y-axis. It is the value of y when x is equal to 0. In other words, it is the point on the line where the x-coordinate is 0.
Q: How do I find the y-intercept of a linear equation?
A: To find the y-intercept of a linear equation, you can use the formula:
b = y - mx
where m is the slope of the line and x is the x-coordinate of the point at which the line intersects the y-axis.
Q: How do I express the y-intercept as a coordinate pair?
A: To express the y-intercept as a coordinate pair, you can write it as (0, b). This means that the y-intercept is the point (0, b) on the coordinate plane.
Q: What is the difference between the slope and y-intercept?
A: The slope and y-intercept are two different aspects of a linear equation. The slope measures how steep the line is, while the y-intercept measures the point at which the line intersects the y-axis.
Q: How do I use the slope and y-intercept to graph a linear equation?
A: To graph a linear equation, you can use the slope and y-intercept to find two points on the line. Then, you can draw a line through those points to graph the equation.
Q: Can I have a negative slope?
A: Yes, you can have a negative slope. A negative slope means that the line falls as you move along the x-axis.
Q: Can I have a zero slope?
A: Yes, you can have a zero slope. A zero slope means that the line is horizontal and does not rise or fall as you move along the x-axis.
Q: Can I have a vertical line?
A: Yes, you can have a vertical line. A vertical line has an undefined slope and is represented by the equation x = a, where a is a constant.
Q: How do I find the equation of a line given two points?
A: To find the equation of a line given two points, you can use the slope formula to find the slope of the line. Then, you can use the point-slope form of a linear equation to write the equation of the line.
Q: How do I find the equation of a line given the slope and y-intercept?
A: To find the equation of a line given the slope and y-intercept, you can use the slope-intercept form of a linear equation to write the equation of the line.
Conclusion
In this article, we answered some frequently asked questions about finding the slope and y-intercept of a linear equation. We covered topics such as the definition of a linear equation, the slope and y-intercept, and how to use the slope and y-intercept to graph a linear equation. We also answered questions about negative slopes, zero slopes, vertical lines, and how to find the equation of a line given two points or the slope and y-intercept.