For Which Independent Value Do The Equations Generate The Same Dependent Value?$\[ \begin{array}{l} y_1 = 6x - 16 \\ y_2 = 3x - 10 \end{array} \\]a. 2 B. 3 C. -2 D. None Please Select The Best Answer From The Choices Provided:A B C D
Introduction
In mathematics, linear equations are a fundamental concept used to model real-world problems. These equations are in the form of y = mx + b, where m is the slope and b is the y-intercept. When we have two linear equations, we can solve them to find the values of the variables. In this article, we will explore how to find the independent value for which the two equations generate the same dependent value.
Understanding the Problem
We are given two linear equations:
y1 = 6x - 16 y2 = 3x - 10
Our goal is to find the value of x for which both equations produce the same value of y.
Setting Up the Equations
To find the value of x, we need to set the two equations equal to each other. This is because we want to find the value of x for which both equations produce the same value of y.
6x - 16 = 3x - 10
Solving for x
Now that we have set up the equation, we can solve for x. To do this, we need to isolate the variable x on one side of the equation.
First, let's add 16 to both sides of the equation:
6x = 3x + 6
Next, let's subtract 3x from both sides of the equation:
3x = 6
Finally, let's divide both sides of the equation by 3:
x = 2
Conclusion
In this article, we have solved two linear equations to find the value of x for which both equations produce the same value of y. We have used algebraic manipulation to isolate the variable x and have found that x = 2.
Answer
The correct answer is:
A. 2
Why is this the correct answer?
This is the correct answer because when we substitute x = 2 into both equations, we get the same value of y.
For the first equation, y1 = 6(2) - 16 = 12 - 16 = -4
For the second equation, y2 = 3(2) - 10 = 6 - 10 = -4
As we can see, both equations produce the same value of y, which is -4.
What does this mean?
This means that when x = 2, both equations produce the same value of y. This is the independent value for which the two equations generate the same dependent value.
Real-World Applications
This concept has many real-world applications. For example, in physics, we can use linear equations to model the motion of objects. In economics, we can use linear equations to model the relationship between variables such as supply and demand.
Conclusion
In conclusion, we have solved two linear equations to find the value of x for which both equations produce the same value of y. We have used algebraic manipulation to isolate the variable x and have found that x = 2. This concept has many real-world applications and is an important tool in mathematics and science.
References
- [1] "Linear Equations" by Math Open Reference
- [2] "Solving Linear Equations" by Khan Academy
Additional Resources
- [1] "Linear Equations" by Wolfram MathWorld
- [2] "Solving Linear Equations" by MIT OpenCourseWare
Frequently Asked Questions: Solving 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. It is typically written in the form of y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
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. For example, y = 2x + 3 is a linear equation, while y = x^2 + 2x + 1 is a quadratic equation.
Q: How do I find the value of x for which two linear equations produce the same value of y?
A: To find the value of x for which two linear equations produce the same value of y, you need to set the two equations equal to each other and solve for x. This is done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
Q: What is the concept of slope in linear equations?
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. The slope is represented by the letter m in the equation y = mx + b.
Q: How do I graph a linear equation?
A: To graph a linear equation, you need to plot two points on the graph and draw a line through them. You can find the x-intercept and y-intercept of the line by setting x = 0 and y = 0, respectively.
Q: What is the concept of y-intercept in linear equations?
A: The y-intercept of a linear equation is the point at which the line crosses the y-axis. It is represented by the letter b in the equation y = mx + b.
Q: How do I use linear equations in real-world applications?
A: Linear equations are used in many real-world applications, such as physics, engineering, economics, and finance. They are used to model the motion of objects, the relationship between variables, and the behavior of systems.
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not isolating the variable on one side of the equation
- Not using the correct order of operations
- Not checking the solution for extraneous solutions
- Not using the correct method for solving the equation
Q: How do I check my solution for extraneous solutions?
A: To check your solution for extraneous solutions, you need to plug the solution back into the original equation and check if it is true. If it is not true, then the solution is extraneous and should be discarded.
Q: What are some common applications of linear equations?
A: Some common applications of linear equations include:
- Modeling the motion of objects
- Modeling the relationship between variables
- Modeling the behavior of systems
- Solving problems in physics, engineering, economics, and finance
Q: How do I use technology to solve linear equations?
A: There are many software programs and online tools available that can help you solve linear equations, such as graphing calculators, computer algebra systems, and online equation solvers.