What Is The Solution To The System Of Equations Given Below?${ \begin{array}{l} y = 3x + 11 \ y = 2x + 12 \end{array} }$
Introduction
System of Equations is a set of two or more equations that contain the same variables. In this article, we will focus on solving a system of two linear equations with two variables. The system of equations given below is a classic example of a linear system.
{ \begin{array}{l} y = 3x + 11 \\ y = 2x + 12 \end{array} \}
Understanding the Problem
The problem is to find the values of x and y that satisfy both equations simultaneously. In other words, we need to find the point of intersection of the two lines represented by the equations.
What are Linear Equations?
Linear equations are equations in which the highest power of the variable(s) is 1. In this case, both equations are linear equations in two variables, x and y.
What is a System of Equations?
A system of equations is a set of two or more equations that contain the same variables. In this case, we have two linear equations with two variables.
Solving the System of Equations
To solve the system of equations, we can use the method of substitution or elimination. In this case, we will use the method of substitution.
Method of Substitution
The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.
Step 1: Solve the First Equation for y
We will solve the first equation for y:
y = 3x + 11
Step 2: Substitute the Expression for y into the Second Equation
We will substitute the expression for y into the second equation:
y = 2x + 12
Substituting y = 3x + 11 into the second equation, we get:
3x + 11 = 2x + 12
Step 3: Solve for x
We will solve for x:
3x + 11 = 2x + 12
Subtracting 2x from both sides, we get:
x + 11 = 12
Subtracting 11 from both sides, we get:
x = 1
Step 4: Find the Value of y
Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. We will use the first equation:
y = 3x + 11
Substituting x = 1, we get:
y = 3(1) + 11
y = 3 + 11
y = 14
Conclusion
The solution to the system of equations is x = 1 and y = 14.
Graphical Method
Another way to solve the system of equations is by graphing the two lines on a coordinate plane. The point of intersection of the two lines represents the solution to the system.
Step 1: Graph the First Line
We will graph the first line, y = 3x + 11.
Step 2: Graph the Second Line
We will graph the second line, y = 2x + 12.
Step 3: Find the Point of Intersection
The point of intersection of the two lines represents the solution to the system.
Conclusion
In this article, we have discussed the solution to the system of equations given below:
{ \begin{array}{l} y = 3x + 11 \\ y = 2x + 12 \end{array} \}
We have used the method of substitution to solve the system of equations and found the solution to be x = 1 and y = 14. We have also discussed the graphical method of solving the system of equations.
Frequently Asked Questions
Q: What is a system of equations?
A: A system of equations is a set of two or more equations that contain the same variables.
Q: What is the method of substitution?
A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.
Q: How do I graph a line on a coordinate plane?
A: To graph a line on a coordinate plane, you need to plot two points on the line and then draw a line through the points.
Q: What is the point of intersection of two lines?
A: The point of intersection of two lines represents the solution to the system of equations.
References
- [1] "System of Equations" by Math Open Reference
- [2] "Method of Substitution" by Khan Academy
- [3] "Graphing Lines" by Math Is Fun
Conclusion
In conclusion, solving a system of equations involves finding the values of the variables that satisfy both equations simultaneously. We have discussed the method of substitution and the graphical method of solving the system of equations. We have also provided some frequently asked questions and references for further reading.
Introduction
Solving systems of equations can be a challenging task, especially for those who are new to the concept. In this article, we will address some of the most frequently asked questions about systems of equations.
Q: What is a system of equations?
A: A system of equations is a set of two or more equations that contain the same variables. In other words, it is a collection of equations that need to be solved simultaneously.
Example:
{ \begin{array}{l} y = 3x + 11 \\ y = 2x + 12 \end{array} \}
This is an example of a system of two linear equations with two variables.
Q: How do I solve a system of equations?
A: There are several methods to solve a system of equations, including the method of substitution, the method of elimination, and the graphical method.
Method of Substitution:
The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.
Method of Elimination:
The method of elimination involves adding or subtracting the equations to eliminate one of the variables.
Graphical Method:
The graphical method involves graphing the two lines on a coordinate plane and finding the point of intersection.
Q: What is the difference between a system of equations and a single equation?
A: A single equation is a statement that says that two expressions are equal. A system of equations is a collection of two or more equations that need to be solved simultaneously.
Example:
y = 3x + 11
This is a single equation. It says that y is equal to 3x + 11.
{ \begin{array}{l} y = 3x + 11 \\ y = 2x + 12 \end{array} \}
This is a system of two linear equations with two variables.
Q: Can I use a calculator to solve a system of equations?
A: Yes, you can use a calculator to solve a system of equations. Many calculators have built-in functions for solving systems of equations.
Example:
Using a calculator to solve the system of equations:
{ \begin{array}{l} y = 3x + 11 \\ y = 2x + 12 \end{array} \}
You can enter the two equations into the calculator and press the "solve" button to find the solution.
Q: What is the point of intersection of two lines?
A: The point of intersection of two lines represents the solution to the system of equations.
Example:
The point of intersection of the two lines y = 3x + 11 and y = 2x + 12 is (1, 14).
Q: Can I graph a system of equations on a coordinate plane?
A: Yes, you can graph a system of equations on a coordinate plane. This is known as the graphical method of solving systems of equations.
Example:
Graphing the system of equations:
{ \begin{array}{l} y = 3x + 11 \\ y = 2x + 12 \end{array} \}
You can plot the two lines on a coordinate plane and find the point of intersection.
Q: What is the difference between a linear system and a nonlinear system?
A: A linear system is a system of equations in which the highest power of the variable(s) is 1. A nonlinear system is a system of equations in which the highest power of the variable(s) is greater than 1.
Example:
The system of equations:
{ \begin{array}{l} y = 3x + 11 \\ y = 2x + 12 \end{array} \}
is a linear system.
The system of equations:
{ \begin{array}{l} y = x^2 + 11 \\ y = 2x + 12 \end{array} \}
is a nonlinear system.
Q: Can I use a computer program to solve a system of equations?
A: Yes, you can use a computer program to solve a system of equations. Many computer programs have built-in functions for solving systems of equations.
Example:
Using a computer program to solve the system of equations:
{ \begin{array}{l} y = 3x + 11 \\ y = 2x + 12 \end{array} \}
You can enter the two equations into the computer program and press the "solve" button to find the solution.
Conclusion
In conclusion, solving systems of equations can be a challenging task, but with the right tools and techniques, it can be done. We have addressed some of the most frequently asked questions about systems of equations and provided examples to illustrate the concepts.
References
- [1] "System of Equations" by Math Open Reference
- [2] "Method of Substitution" by Khan Academy
- [3] "Graphing Lines" by Math Is Fun
Additional Resources
- [1] "Solving Systems of Equations" by Mathway
- [2] "Systems of Equations" by Wolfram Alpha
- [3] "Solving Systems of Equations" by MIT OpenCourseWare