Solve For $y$: 2 Y − 8 X = 20 Y = [ ? ] X + □ \begin{array}{l} 2y - 8x = 20 \\ y = [?]x + \square \end{array} 2 Y − 8 X = 20 Y = [ ?] X + □
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill to master. In this article, we will focus on solving a specific type of linear equation, where the variable y is expressed in terms of x. We will use the given equation 2y - 8x = 20 to solve for y and express it in the form y = mx + b, where m is the slope and b is the y-intercept.
Understanding the Equation
The given equation is 2y - 8x = 20. To solve for y, we need to isolate the variable y on one side of the equation. We can start by adding 8x to both sides of the equation to get:
2y = 20 + 8x
Simplifying the Equation
Next, we can simplify the equation by dividing both sides by 2 to get:
y = (20 + 8x) / 2
Simplifying Further
We can simplify the equation further by combining the constants on the right-hand side:
y = 10 + 4x
Expressing y in Terms of x
Now that we have simplified the equation, we can express y in terms of x. The equation is now in the form y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope (m) is 4, and the y-intercept (b) is 10.
Interpreting the Results
The equation y = 4x + 10 represents a linear relationship between x and y. The slope of 4 means that for every unit increase in x, y increases by 4 units. The y-intercept of 10 means that when x is equal to 0, y is equal to 10.
Real-World Applications
Linear equations have many real-world applications, including:
- Physics: Linear equations are used to describe the motion of objects under constant acceleration.
- Economics: Linear equations are used to model the relationship between variables such as supply and demand.
- Computer Science: Linear equations are used in algorithms such as linear regression to model the relationship between variables.
Conclusion
Solving linear equations is a crucial skill to master in mathematics. By following the steps outlined in this article, we can solve for y in the equation 2y - 8x = 20 and express it in the form y = mx + b. The equation y = 4x + 10 represents a linear relationship between x and y, and has many real-world applications.
Additional Resources
For more information on solving linear equations, check out the following resources:
- Mathway: A online math problem solver that can help you solve linear equations.
- Khan Academy: A free online resource that provides video lessons and practice exercises on solving linear equations.
- MIT OpenCourseWare: A free online resource that provides lecture notes and practice exercises on solving linear equations.
Frequently Asked Questions
Q: What is the slope of the equation y = 4x + 10?
A: The slope of the equation y = 4x + 10 is 4.
Q: What is the y-intercept of the equation y = 4x + 10?
A: The y-intercept of the equation y = 4x + 10 is 10.
Q: How do I solve a linear equation with two variables?
Introduction
Solving linear equations is a crucial skill to master in mathematics. In our previous article, we provided a step-by-step guide on solving a specific type of linear equation, where the variable y is expressed in terms of x. In this article, we will provide a Q&A guide to help you better understand and solve linear equations.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants.
Q: How do I solve a linear equation?
A: To solve a linear equation, you can use the following steps:
- Isolate the variable: Move all the terms with the variable to one side of the equation.
- Combine like terms: Combine any like terms on the same side of the equation.
- Solve for the variable: Solve for the variable by dividing or multiplying both sides of the equation by the coefficient of the variable.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. In other words, a linear equation can be written in the form ax + b = c, while a quadratic equation can be written in the form ax^2 + bx + c = 0.
Q: How do I graph a linear equation?
A: To graph a linear equation, you can use the following steps:
- Find the x-intercept: Find the point where the line crosses the x-axis by setting y = 0 and solving for x.
- Find the y-intercept: Find the point where the line crosses the y-axis by setting x = 0 and solving for y.
- Plot the points: Plot the x-intercept and y-intercept on a coordinate plane.
- Draw the line: Draw a line through the two points to represent the linear equation.
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 by dividing the change in y by the change in x. In other words, it is calculated by:
m = (y2 - y1) / (x2 - x1)
Q: How do I find the slope of a linear equation?
A: To find the slope of a linear equation, you can use the following steps:
- Find two points on the line: Find two points on the line by plotting the x-intercept and y-intercept.
- Calculate the change in y: Calculate the change in y by subtracting the y-coordinates of the two points.
- Calculate the change in x: Calculate the change in x by subtracting the x-coordinates of the two points.
- Calculate the slope: Calculate the slope by dividing the change in y by the change in x.
Q: What is the y-intercept of a linear equation?
A: The y-intercept of a linear equation is the point where the line crosses the y-axis. It is calculated by setting x = 0 and solving for y.
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 following steps:
- Set x = 0: Set x = 0 in the equation.
- Solve for y: Solve for y by dividing or multiplying both sides of the equation by the coefficient of the variable.
Conclusion
Solving linear equations is a crucial skill to master in mathematics. By following the steps outlined in this article, you can solve linear equations and understand the concepts of slope and y-intercept. We hope this Q&A guide has been helpful in answering your questions and providing a better understanding of linear equations.
Additional Resources
For more information on solving linear equations, check out the following resources:
- Mathway: A online math problem solver that can help you solve linear equations.
- Khan Academy: A free online resource that provides video lessons and practice exercises on solving linear equations.
- MIT OpenCourseWare: A free online resource that provides lecture notes and practice exercises on solving linear equations.
Frequently Asked Questions
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2.
Q: How do I graph a linear equation?
A: To graph a linear equation, you can use the following steps: find the x-intercept, find the y-intercept, plot the points, and draw the line.
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 by dividing the change in y by the change in x.
Q: How do I find the slope of a linear equation?
A: To find the slope of a linear equation, you can use the following steps: find two points on the line, calculate the change in y, calculate the change in x, and calculate the slope.