Given The Equation $y = 42x + 900$, Find The Slope And The $y$-intercept.
Introduction
In mathematics, linear equations are a fundamental concept that describe a relationship between two variables, typically represented as x and y. The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. In this article, we will explore how to find the slope and y-intercept of a linear equation given the equation y = 42x + 900.
Understanding the Components of a Linear Equation
A linear equation is composed of two main components: the slope (m) and the y-intercept (b). The slope represents the rate of change of the dependent variable (y) with respect to the independent variable (x). It is a measure of how steep the line is. The y-intercept, on the other hand, is the point at which the line intersects the y-axis. It represents the value of y when x is equal to zero.
Finding the Slope
The slope of a linear equation is represented by the coefficient of the x term. In the given equation y = 42x + 900, the coefficient of the x term is 42. Therefore, the slope of the equation is 42.
Finding the Y-Intercept
The y-intercept of a linear equation is the value of y when x is equal to zero. In the given equation y = 42x + 900, we can find the y-intercept by substituting x = 0 into the equation.
y = 42(0) + 900 y = 0 + 900 y = 900
Therefore, the y-intercept of the equation is 900.
Interpretation of Results
Now that we have found the slope and y-intercept of the equation, we can interpret the results. The slope of 42 indicates that for every unit increase in x, y increases by 42 units. The y-intercept of 900 indicates that the line intersects the y-axis at the point (0, 900).
Real-World Applications
The concept of slope and y-intercept has numerous real-world applications. In economics, the slope of a demand curve represents the rate of change of the quantity demanded with respect to the price. In physics, the slope of a velocity-time graph represents the acceleration of an object. In engineering, the slope of a stress-strain graph represents the modulus of elasticity of a material.
Conclusion
In conclusion, finding the slope and y-intercept of a linear equation is a fundamental concept in mathematics. By understanding the components of a linear equation and applying the necessary formulas, we can determine the slope and y-intercept of any given equation. The concept of slope and y-intercept has numerous real-world applications and is an essential tool for problem-solving in various fields.
Example Problems
- Find the slope and y-intercept of the equation y = 3x + 200.
- Find the slope and y-intercept of the equation y = 2x - 50.
- Find the slope and y-intercept of the equation y = 5x + 300.
Solutions
- The slope of the equation is 3, and the y-intercept is 200.
- The slope of the equation is 2, and the y-intercept is -50.
- The slope of the equation is 5, and the y-intercept is 300.
Tips and Tricks
- When finding the slope, make sure to identify the coefficient of the x term.
- When finding the y-intercept, substitute x = 0 into the equation.
- Use the slope-intercept form of a linear equation (y = mx + b) to find the slope and y-intercept.
Frequently Asked Questions
- What is the slope of a linear equation?
- What is the y-intercept of a linear equation?
- How do I find the slope and y-intercept of a linear equation?
Answers
- The slope of a linear equation is the coefficient of the x term.
- The y-intercept of a linear equation is the value of y when x is equal to zero.
- To find the slope and y-intercept of a linear equation, use the slope-intercept form of a linear equation (y = mx + b) and substitute x = 0 into the equation.
Frequently Asked Questions About Slope and Y-Intercept ===========================================================
Q: What is the slope of a linear equation?
A: The slope of a linear equation is the coefficient of the x term. It represents the rate of change of the dependent variable (y) with respect to the independent variable (x).
Q: What is the y-intercept of a linear equation?
A: The y-intercept of a linear equation is the value of y when x is equal to zero. It represents 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, use the slope-intercept form of a linear equation (y = mx + b) and substitute x = 0 into the equation. The slope is the coefficient of the x term, and the y-intercept is the value of y when x is equal to zero.
Q: What is the difference between slope and y-intercept?
A: The slope represents the rate of change of the dependent variable (y) with respect to the independent variable (x), while the y-intercept represents 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, use the slope-intercept form of a linear equation (y = mx + b) and plot the y-intercept on the y-axis. Then, use the slope to determine the direction and steepness of the line.
Q: What are some real-world applications of slope and y-intercept?
A: The concept of slope and y-intercept has numerous real-world applications, including economics, physics, and engineering. In economics, the slope of a demand curve represents the rate of change of the quantity demanded with respect to the price. In physics, the slope of a velocity-time graph represents the acceleration of an object. In engineering, the slope of a stress-strain graph represents the modulus of elasticity of a material.
Q: How do I determine the equation of a line given the slope and y-intercept?
A: To determine the equation of a line given the slope and y-intercept, use the slope-intercept form of a linear equation (y = mx + b) and substitute the slope and y-intercept into the equation.
Q: What is the equation of a line with a slope of 2 and a y-intercept of 3?
A: The equation of a line with a slope of 2 and a y-intercept of 3 is y = 2x + 3.
Q: What is the equation of a line with a slope of -1 and a y-intercept of 2?
A: The equation of a line with a slope of -1 and a y-intercept of 2 is y = -x + 2.
Q: How do I find the equation of a line given two points?
A: To find the equation of a line given two points, use the slope formula (m = (y2 - y1) / (x2 - x1)) to determine the slope, and then use the point-slope form of a linear equation (y - y1 = m(x - x1)) to determine the equation of the line.
Q: What is the equation of a line passing through the points (1, 2) and (3, 4)?
A: The equation of a line passing through the points (1, 2) and (3, 4) is y = (4/2)(x - 1) + 2, which simplifies to y = 2x + 1.
Q: How do I use the equation of a line to solve problems?
A: To use the equation of a line to solve problems, substitute the given values into the equation and solve for the unknown variable.
Q: What are some common mistakes to avoid when working with slope and y-intercept?
A: Some common mistakes to avoid when working with slope and y-intercept include:
- Confusing the slope and y-intercept
- Failing to use the correct formula for the slope
- Failing to substitute the correct values into the equation
- Failing to simplify the equation
Q: How do I check my work when working with slope and y-intercept?
A: To check your work when working with slope and y-intercept, use the following steps:
- Verify that the slope and y-intercept are correct
- Verify that the equation of the line is correct
- Use the equation of the line to solve a problem and verify that the solution is correct