Select The Correct Answer.Suppose The Following Function Is Graphed: $ Y = \frac{8}{5}x + 4 }$On The Same Grid, A New Function Is Graphed. The New Function Is Represented By The Following Equation ${ Y = -\frac{5 {8}x + 8 }$Which
Introduction
In mathematics, the intersection of two lines is a fundamental concept that helps us understand the relationship between different linear functions. When two lines intersect, they share a common point, and this point is known as the intersection point. In this article, we will explore the concept of the intersection of two linear functions and how to find the correct answer when given two equations.
What are Linear Functions?
A linear function is a function that can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. The slope of a linear function represents the rate of change of the function, while the y-intercept represents the point where the function intersects the y-axis.
The Given Functions
The two functions given in the problem are:
- y = (8/5)x + 4
- y = (-5/8)x + 8
Understanding the Slope and Y-Intercept
The slope of the first function is 8/5, which means that for every unit increase in x, the value of y increases by 8/5 units. The y-intercept of the first function is 4, which means that when x is equal to 0, the value of y is 4.
The slope of the second function is -5/8, which means that for every unit increase in x, the value of y decreases by 5/8 units. The y-intercept of the second function is 8, which means that when x is equal to 0, the value of y is 8.
Finding the Intersection Point
To find the intersection point of the two functions, we need to set the two equations equal to each other and solve for x.
(8/5)x + 4 = (-5/8)x + 8
To solve for x, we can multiply both sides of the equation by the least common multiple of 5 and 8, which is 40.
40((8/5)x + 4) = 40((-5/8)x + 8)
This simplifies to:
32x + 160 = -25x + 320
Now, we can add 25x to both sides of the equation to get:
57x + 160 = 320
Next, we can subtract 160 from both sides of the equation to get:
57x = 160
Finally, we can divide both sides of the equation by 57 to get:
x = 160/57
x = 2.81
Finding the Y-Coordinate
Now that we have found the x-coordinate of the intersection point, we can substitute this value into one of the original equations to find the y-coordinate.
Let's use the first equation:
y = (8/5)x + 4
y = (8/5)(2.81) + 4
y = 4.53 + 4
y = 8.53
Conclusion
In conclusion, the intersection point of the two functions is (2.81, 8.53). This means that the two lines intersect at a point that is approximately 2.81 units to the right of the y-axis and 8.53 units above the x-axis.
Why is this Important?
Understanding the intersection of two linear functions is important in many real-world applications, such as:
- Physics: The intersection of two lines can represent the trajectory of an object under the influence of gravity.
- Engineering: The intersection of two lines can represent the stress and strain on a material under different loads.
- Economics: The intersection of two lines can represent the equilibrium price and quantity of a good in a market.
Final Thoughts
Q: What is the intersection of two linear functions?
A: The intersection of two linear functions is the point where the two lines intersect. This point is also known as the solution to the system of equations.
Q: How do I find the intersection point of two linear functions?
A: To find the intersection point of two linear functions, you need to set the two equations equal to each other and solve for x. Once you have found the x-coordinate of the intersection point, you can substitute this value into one of the original equations to find the y-coordinate.
Q: What if the two lines are parallel?
A: If the two lines are parallel, they will never intersect. This means that there is no solution to the system of equations.
Q: What if the two lines are perpendicular?
A: If the two lines are perpendicular, they will intersect at a single point. This point is also known as the solution to the system of equations.
Q: How do I know if two lines are parallel or perpendicular?
A: To determine if two lines are parallel or perpendicular, you need to compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.
Q: What is the significance of the intersection point of two linear functions?
A: The intersection point of two linear functions is significant because it represents the solution to the system of equations. This solution can be used to make predictions, model real-world phenomena, and solve problems in various fields such as physics, engineering, and economics.
Q: Can I use the intersection point of two linear functions to solve real-world problems?
A: Yes, the intersection point of two linear functions can be used to solve real-world problems. For example, in physics, the intersection point of two lines can represent the trajectory of an object under the influence of gravity. In engineering, the intersection point of two lines can represent the stress and strain on a material under different loads.
Q: What are some common applications of the intersection point of two linear functions?
A: Some common applications of the intersection point of two linear functions include:
- Physics: The intersection point of two lines can represent the trajectory of an object under the influence of gravity.
- Engineering: The intersection point of two lines can represent the stress and strain on a material under different loads.
- Economics: The intersection point of two lines can represent the equilibrium price and quantity of a good in a market.
- Computer Science: The intersection point of two lines can be used to solve problems in computer graphics, game development, and computer vision.
Q: How do I graph the intersection point of two linear functions?
A: To graph the intersection point of two linear functions, you need to plot the two lines on a coordinate plane and find the point where they intersect. This point is the solution to the system of equations.
Q: What are some common mistakes to avoid when finding the intersection point of two linear functions?
A: Some common mistakes to avoid when finding the intersection point of two linear functions include:
- Not setting the two equations equal to each other.
- Not solving for x correctly.
- Not substituting the x-coordinate into one of the original equations to find the y-coordinate.
- Not checking for parallel or perpendicular lines.
Q: How do I check if two lines are parallel or perpendicular?
A: To check if two lines are parallel or perpendicular, you need to compare their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.
Q: What are some real-world examples of the intersection point of two linear functions?
A: Some real-world examples of the intersection point of two linear functions include:
- The trajectory of a projectile under the influence of gravity.
- The stress and strain on a material under different loads.
- The equilibrium price and quantity of a good in a market.
- The solution to a system of equations in computer graphics, game development, and computer vision.