What Is The Solution To This System Of Equations?${ \begin{array}{l} y = \frac{2}{3} X - 1 \ y = 3x + 5 \end{array} }$
Introduction
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. In this article, we will focus on solving a system of two linear equations with two variables. The system of equations we will be solving is given by:
{ \begin{array}{l} y = \frac{2}{3} x - 1 \\ y = 3x + 5 \end{array} \}
Understanding the System of Equations
The system of equations consists of two linear equations with two variables, x and y. The first equation is , and the second equation is . To solve this system of equations, we need to find the values of x and y that satisfy both equations simultaneously.
Method of Substitution
One of the methods to solve a system of equations is the method of substitution. This method involves solving one of the equations for one variable and then substituting that expression into the other equation. In this case, we can solve the first equation for y and then substitute that expression into the second equation.
Step 1: Solve the First Equation for y
To solve the first equation for y, we can isolate y on one side of the equation. We can do this by adding 1 to both sides of the equation:
{ y = \frac{2}{3} x - 1 \}
Adding 1 to both sides:
{ y + 1 = \frac{2}{3} x \}
Subtracting 1 from both sides:
{ y = \frac{2}{3} x - 1 \}
Step 2: Substitute the Expression for y into the Second Equation
Now that we have solved the first equation for y, we can substitute that expression into the second equation:
{ y = 3x + 5 \}
Substituting the expression for y:
{ \frac{2}{3} x - 1 = 3x + 5 \}
Solving the Resulting Equation
Now that we have substituted the expression for y into the second equation, we can solve the resulting equation for x. To do this, we can add 1 to both sides of the equation:
{ \frac{2}{3} x - 1 + 1 = 3x + 5 + 1 \}
Simplifying the equation:
{ \frac{2}{3} x = 3x + 6 \}
Subtracting 3x from both sides:
{ -\frac{1}{3} x = 6 \}
Multiplying both sides by -3:
{ x = -18 \}
Finding the Value of y
Now that we have found the value of x, we can substitute that value into one of the original equations to find the value of y. We can use the first equation:
{ y = \frac{2}{3} x - 1 \}
Substituting the value of x:
{ y = \frac{2}{3} (-18) - 1 \}
Simplifying the equation:
{ y = -12 - 1 \}
{ y = -13 \}
Conclusion
In this article, we have solved a system of two linear equations with two variables using the method of substitution. We have found the values of x and y that satisfy both equations simultaneously. The value of x is -18, and the value of y is -13. This solution satisfies both equations, and it is the only solution to the system of equations.
Final Answer
The final answer to the system of equations is:
{ x = -18 \}
{ y = -13 \}
This solution satisfies both equations, and it is the only solution to the system of equations.
Introduction
Solving systems of equations can be a challenging task, especially for those who are new to algebra. However, with the right approach and techniques, it can be a fun and rewarding experience. In this article, we will answer some of the most frequently asked questions about solving systems of equations.
Q: What is a system of equations?
A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In other words, it is a collection of equations that are related to each other and must be solved together.
Q: What are the different methods for solving systems of equations?
A: There are several methods for solving systems of equations, including:
- Method of Substitution: This method involves solving one of the equations for one variable and then substituting that expression into the other equation.
- Method of Elimination: This method involves adding or subtracting the equations to eliminate one of the variables.
- Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
Q: What is the difference between a linear equation and a nonlinear equation?
A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 5 is a linear equation. A nonlinear equation, on the other hand, is an equation in which the highest power of the variable is greater than 1. For example, x^2 + 2x + 1 = 0 is a nonlinear equation.
Q: Can a system of equations have more than one solution?
A: Yes, a system of equations can have more than one solution. This is known as an inconsistent system. For example, the system of equations:
{ \begin{array}{l} x + y = 2 \\ x + y = 3 \end{array} \}
has no solution, because the two equations are contradictory.
Q: Can a system of equations have no solution?
A: Yes, a system of equations can have no solution. This is known as an inconsistent system. For example, the system of equations:
{ \begin{array}{l} x + y = 2 \\ x + y = 3 \end{array} \}
has no solution, because the two equations are contradictory.
Q: Can a system of equations have infinitely many solutions?
A: Yes, a system of equations can have infinitely many solutions. This is known as a dependent system. For example, the system of equations:
{ \begin{array}{l} x + y = 2 \\ y = x + 1 \end{array} \}
has infinitely many solutions, because the two equations are equivalent.
Q: How do I know which method to use to solve a system of equations?
A: The choice of method depends on the type of equations and the number of variables. If the equations are linear and there are two variables, the method of substitution or elimination may be used. If the equations are nonlinear or there are more than two variables, the graphical method or other advanced techniques may be used.
Q: What are some common mistakes to avoid when solving systems of equations?
A: Some common mistakes to avoid when solving systems of equations include:
- Not checking for extraneous solutions: Make sure to check that the solution satisfies both equations.
- Not using the correct method: Choose the method that is best suited for the type of equations and the number of variables.
- Not simplifying the equations: Simplify the equations before solving them to make it easier to find the solution.
Conclusion
Solving systems of equations can be a challenging task, but with the right approach and techniques, it can be a fun and rewarding experience. By understanding the different methods for solving systems of equations and avoiding common mistakes, you can become proficient in solving systems of equations and apply this skill to a wide range of problems.