Solve The System: $\[ \begin{cases} y - 10 = 3x \\ 2y = 6x + 20 \end{cases} \\]A. No SolutionB. Infinite SolutionsC. (10, 40)D. (2, 13)
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Introduction
Solving a system of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the solution to the system.
The System of Equations
The given system of equations is:
{ \begin{cases} y - 10 = 3x \\ 2y = 6x + 20 \end{cases} \}
Step 1: Solve the First Equation for y
To solve the system, we can start by solving the first equation for y. We can add 10 to both sides of the equation to isolate y.
{ y - 10 + 10 = 3x + 10 \}
{ y = 3x + 10 \}
Step 2: Substitute the Expression for y into the Second Equation
Now that we have an expression for y, we can substitute it into the second equation.
{ 2(3x + 10) = 6x + 20 \}
Step 3: Expand and Simplify the Equation
We can expand and simplify the equation by multiplying the 2 to the terms inside the parentheses.
{ 6x + 20 = 6x + 20 \}
Step 4: Analyze the Result
The equation is a true statement, which means that the two equations are equivalent. This implies that the system of equations has infinite solutions.
Conclusion
In conclusion, the system of equations has infinite solutions. This means that there are an infinite number of points that satisfy both equations.
Final Answer
The final answer is:
- B. Infinite solutions
Discussion
The system of equations has infinite solutions because the two equations are equivalent. This can be seen by substituting the expression for y into the second equation, which results in a true statement. This means that any point that satisfies the first equation will also satisfy the second equation, and vice versa.
Example
To illustrate this concept, let's consider an example. Suppose we have the system of equations:
{ \begin{cases} y - 5 = 2x \\ y = 3x + 5 \end{cases} \}
We can solve the first equation for y and substitute it into the second equation.
{ y - 5 + 5 = 2x + 5 \}
{ y = 2x + 5 \}
Substituting this expression into the second equation, we get:
{ 2x + 5 = 3x + 5 \}
This equation is also true, which means that the system of equations has infinite solutions.
Tips and Tricks
When solving a system of equations, it's essential to check if the two equations are equivalent. If they are, then the system has infinite solutions. On the other hand, if the two equations are not equivalent, then the system may have a unique solution or no solution at all.
Common Mistakes
One common mistake when solving a system of equations is to assume that the system has a unique solution when it actually has infinite solutions. This can happen when the two equations are equivalent, but the solution is not immediately apparent.
Real-World Applications
Solving a system of equations has numerous real-world applications. For example, in physics, we can use systems of equations to model the motion of objects. In engineering, we can use systems of equations to design and optimize systems. In economics, we can use systems of equations to model the behavior of markets.
Conclusion
In conclusion, solving a system of equations is a fundamental concept in mathematics that has numerous applications in various fields. By using the method of substitution and elimination, we can solve systems of equations and determine if they have a unique solution, no solution, or infinite solutions.
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Frequently Asked Questions
Q: What is a system of equations?
A: A system of equations is a set of two or more equations that are related to each other. Each equation contains one or more variables, and the system is solved by finding the values of the variables that satisfy all the equations.
Q: How do I know if a system of equations has a unique solution, no solution, or infinite solutions?
A: To determine the type of solution, you can use the following methods:
- Substitution method: Solve one equation for one variable and substitute the expression into the other equation.
- Elimination method: Add or subtract the equations to eliminate one variable.
- Graphing method: Graph the equations on a coordinate plane and find the point of intersection.
Q: What is the difference between a system of linear equations and a system of nonlinear equations?
A: A system of linear equations consists of two or more linear equations, while a system of nonlinear equations consists of two or more nonlinear equations. Nonlinear equations are equations that are not linear, meaning they do not have a constant slope.
Q: Can a system of equations have more than two variables?
A: Yes, a system of equations can have more than two variables. However, the number of variables must be equal to the number of equations.
Q: How do I solve a system of equations with more than two variables?
A: To solve a system of equations with more than two variables, you can use the following methods:
- Substitution method: Solve one equation for one variable and substitute the expression into the other equations.
- Elimination method: Add or subtract the equations to eliminate one variable.
- Matrix method: Use matrices to represent the system of equations and solve for the variables.
Q: What is the matrix method for solving systems of equations?
A: The matrix method involves representing the system of equations as a matrix and using row operations to solve for the variables. This method is useful for solving systems of equations with more than two variables.
Q: Can a system of equations have a solution that is not an integer?
A: Yes, a system of equations can have a solution that is not an integer. The solution can be a rational number, a decimal number, or a complex number.
Q: How do I know if a system of equations has a unique solution, no solution, or infinite solutions?
A: To determine the type of solution, you can use the following methods:
- Substitution method: Solve one equation for one variable and substitute the expression into the other equation.
- Elimination method: Add or subtract the equations to eliminate one variable.
- Graphing method: Graph the equations on a coordinate plane and find the point of intersection.
Q: What is the difference between a system of equations and a system of inequalities?
A: A system of equations consists of two or more equations, while a system of inequalities consists of two or more inequalities. Inequalities are statements that compare two expressions using greater than, less than, greater than or equal to, or less than or equal to.
Q: Can a system of inequalities have a solution that is not an integer?
A: Yes, a system of inequalities can have a solution that is not an integer. The solution can be a rational number, a decimal number, or a complex number.
Q: How do I solve a system of inequalities?
A: To solve a system of inequalities, you can use the following methods:
- Graphing method: Graph the inequalities on a coordinate plane and find the region that satisfies all the inequalities.
- Substitution method: Solve one inequality for one variable and substitute the expression into the other inequalities.
- Elimination method: Add or subtract the inequalities to eliminate one variable.
Conclusion
In conclusion, solving systems of equations is a fundamental concept in mathematics that has numerous applications in various fields. By using the substitution method, elimination method, and graphing method, we can solve systems of equations and determine if they have a unique solution, no solution, or infinite solutions. Additionally, we can use the matrix method to solve systems of equations with more than two variables.