Is (3,1) A Solution To This System Of Equations? Y= – 4x+13 Y=2x–5
Introduction
In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In this article, we will be discussing a system of two linear equations in two variables, x and y. The equations are:
- y = –4x + 13
- y = 2x – 5
We will be determining if the point (3,1) is a solution to this system of equations.
What is a Solution to a System of Equations?
A solution to a system of equations is a set of values for the variables that makes all the equations in the system true. In other words, if we substitute the values of the variables into each equation, the equation should be satisfied.
Substituting the Values of the Variables
To determine if the point (3,1) is a solution to the system of equations, we need to substitute the values of x and y into each equation.
Equation 1: y = –4x + 13
Substituting x = 3 and y = 1 into the first equation, we get:
1 = –4(3) + 13
Expanding the equation, we get:
1 = –12 + 13
Simplifying the equation, we get:
1 = 1
This equation is true, which means that the point (3,1) satisfies the first equation.
Equation 2: y = 2x – 5
Substituting x = 3 and y = 1 into the second equation, we get:
1 = 2(3) – 5
Expanding the equation, we get:
1 = 6 – 5
Simplifying the equation, we get:
1 = 1
This equation is also true, which means that the point (3,1) satisfies the second equation.
Conclusion
Since the point (3,1) satisfies both equations in the system, it is a solution to the system of equations.
Graphical Representation
To visualize the solution, we can graph the two equations on a coordinate plane.
Graph of Equation 1: y = –4x + 13
The graph of the first equation is a straight line with a negative slope. The y-intercept is 13, and the x-intercept is –13/4.
Graph of Equation 2: y = 2x – 5
The graph of the second equation is a straight line with a positive slope. The y-intercept is –5, and the x-intercept is 5/2.
Finding the Intersection Point
To find the intersection point of the two lines, we can set the two equations equal to each other and solve for x.
Equating the two equations, we get:
–4x + 13 = 2x – 5
Combine like terms, we get:
–6x = –18
Divide both sides by –6, we get:
x = 3
Substituting x = 3 into one of the equations, we get:
y = 1
Therefore, the intersection point of the two lines is (3,1), which is the solution to the system of equations.
Conclusion
In conclusion, the point (3,1) is a solution to the system of equations y = –4x + 13 and y = 2x – 5. This can be verified by substituting the values of the variables into each equation and checking if the equation is satisfied. The graphical representation of the two equations also confirms that the point (3,1) is the intersection point of the two lines.
Final Answer
The final answer is yes, the point (3,1) is a solution to the system of equations y = –4x + 13 and y = 2x – 5.
Introduction
In the previous article, we discussed a system of two linear equations in two variables, x and y. We determined that the point (3,1) is a solution to the system of equations y = –4x + 13 and y = 2x – 5. In this article, we will be answering some frequently asked questions about systems of equations and solutions.
Q1: What is a system of equations?
A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.
Q2: How do I determine if a point is a solution to a system of equations?
To determine if a point is a solution to a system of equations, you need to substitute the values of the variables into each equation and check if the equation is satisfied.
Q3: What is the difference between a solution and an intersection point?
A solution is a set of values for the variables that makes all the equations in the system true. An intersection point is the point where two or more lines intersect.
Q4: How do I find the intersection point of two lines?
To find the intersection point of two lines, you can set the two equations equal to each other and solve for x. Then, substitute x into one of the equations to find y.
Q5: What if the system of equations has no solution?
If the system of equations has no solution, it means that the two lines are parallel and will never intersect.
Q6: What if the system of equations has infinitely many solutions?
If the system of equations has infinitely many solutions, it means that the two lines are the same and will intersect at every point.
Q7: How do I graph a system of equations?
To graph a system of equations, you can graph each equation separately and then find the intersection point of the two lines.
Q8: What is the importance of solving systems of equations?
Solving systems of equations is important in many real-world applications, such as physics, engineering, and economics.
Q9: Can I use technology to solve systems of equations?
Yes, you can use technology such as calculators or computer software to solve systems of equations.
Q10: What are some common mistakes to avoid when solving systems of equations?
Some common mistakes to avoid when solving systems of equations include:
- Not checking if the system of equations has a solution
- Not finding the intersection point of the two lines
- Not using the correct method to solve the system of equations
Conclusion
In conclusion, solving systems of equations is an important skill that has many real-world applications. By understanding the concepts of solutions and intersection points, you can solve systems of equations with confidence.
Final Answer
The final answer is that solving systems of equations is a crucial skill that requires attention to detail and a thorough understanding of the concepts involved.
Additional Resources
For more information on solving systems of equations, you can refer to the following resources:
- Online tutorials and videos
- Math textbooks and workbooks
- Online math communities and forums
Final Thoughts
Solving systems of equations is a challenging but rewarding topic. By practicing and mastering the concepts, you can become proficient in solving systems of equations and apply them to real-world problems.