Solve The System Of Equations:$\[ \begin{cases} y = \frac{1}{2}x + 3 \\ y = \frac{1}{3}x - 4 \end{cases} \\]

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.

What is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations that involve the same set of variables. Each equation in the system is a linear equation, which means that it can be written in the form:

ax + by = c

where a, b, and c are constants, and x and y are the variables.

Example: A System of Two Linear Equations

Let's consider the following system of two linear equations:

{ \begin{cases} y = \frac{1}{2}x + 3 \\ y = \frac{1}{3}x - 4 \end{cases} \}$ $ In this system, we have two linear equations with two variables, **x** and **y**. Our goal is to find the values of **x** and **y** that satisfy both equations. **Method 1: Substitution Method** --------------------------- One way to solve a system of linear equations is to use the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation. Let's solve the first equation for **y**: **y = \frac{1}{2}x + 3** Now, substitute this expression for **y** into the second equation: **\frac{1}{2}x + 3 = \frac{1}{3}x - 4** To solve for **x**, we can multiply both sides of the equation by 6 to eliminate the fractions: **3x + 18 = 2x - 24** Now, subtract **2x** from both sides of the equation: **x + 18 = -24** Subtract 18 from both sides of the equation: **x = -42** Now that we have found the value of **x**, we can substitute it into one of the original equations to find the value of **y**. Let's use the first equation: **y = \frac{1}{2}x + 3** Substitute **x = -42** into this equation: **y = \frac{1}{2}(-42) + 3** Simplify the expression: **y = -21 + 3** **y = -18** Therefore, the solution to the system of linear equations is **x = -42** and **y = -18**. **Method 2: Elimination Method** --------------------------- Another way to solve a system of linear equations is to use the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables. Let's add the two equations together: **y + y = \frac{1}{2}x + 3 + \frac{1}{3}x - 4** Combine like terms: **2y = \frac{5}{6}x - 1** Now, divide both sides of the equation by 2: **y = \frac{5}{12}x - \frac{1}{2}** Now that we have found the value of **y**, we can substitute it into one of the original equations to find the value of **x**. Let's use the first equation: **y = \frac{1}{2}x + 3** Substitute **y = \frac{5}{12}x - \frac{1}{2}** into this equation: **\frac{5}{12}x - \frac{1}{2} = \frac{1}{2}x + 3** To solve for **x**, we can multiply both sides of the equation by 12 to eliminate the fractions: **5x - 6 = 6x + 36** Now, subtract **6x** from both sides of the equation: **-x - 6 = 36** Add 6 to both sides of the equation: **-x = 42** Multiply both sides of the equation by -1: **x = -42** Now that we have found the value of **x**, we can substitute it into one of the original equations to find the value of **y**. Let's use the first equation: **y = \frac{1}{2}x + 3** Substitute **x = -42** into this equation: **y = \frac{1}{2}(-42) + 3** Simplify the expression: **y = -21 + 3** **y = -18** Therefore, the solution to the system of linear equations is **x = -42** and **y = -18**. **Conclusion** ---------- In this article, we have discussed how to solve a system of linear equations using the substitution method and the elimination method. We have also provided a step-by-step guide on how to solve a system of two linear equations with two variables. By following these steps, you can solve any system of linear equations that you may encounter. **References** -------------- * [1] "Linear Algebra and Its Applications" by Gilbert Strang * [2] "Introduction to Linear Algebra" by Jim Hefferon * [3] "Solving Systems of Linear Equations" by Math Open Reference **Additional Resources** ------------------------- * Khan Academy: Solving Systems of Linear Equations * MIT OpenCourseWare: Linear Algebra * Wolfram MathWorld: Systems of Linear Equations<br/> **Frequently Asked Questions: Solving Systems of Linear Equations** ==================================================================== **Q: What is a system of linear equations?** ----------------------------------------- A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. Each equation in the system is a linear equation, which means that it can be written in the form: **ax + by = c** where **a**, **b**, and **c** are constants, and **x** and **y** are the variables. **Q: How do I know if a system of linear equations has a solution?** --------------------------------------------------------- A: A system of linear equations has a solution if and only if the two equations are consistent, meaning that they do not contradict each other. If the two equations are inconsistent, then the system has no solution. **Q: What are the two main methods for solving systems of linear equations?** ------------------------------------------------------------------- A: The two main methods for solving systems of linear equations are the substitution method and the elimination method. * The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. * The elimination method involves adding or subtracting the equations to eliminate one of the variables. **Q: How do I choose between the substitution method and the elimination method?** ------------------------------------------------------------------------- A: The choice between the substitution method and the elimination method depends on the specific system of linear equations. If one of the equations is already solved for one variable, then the substitution method may be easier to use. If the coefficients of the variables are the same in both equations, then the elimination method may be easier to use. **Q: What if I have a system of linear equations with three or more variables?** ------------------------------------------------------------------------- A: If you have a system of linear equations with three or more variables, then you can use the same methods as before, but you will need to use more variables and more equations. You can also use matrix methods, such as Gaussian elimination or LU decomposition, to solve the system. **Q: Can I use a calculator or computer to solve a system of linear equations?** ------------------------------------------------------------------------- A: Yes, you can use a calculator or computer to solve a system of linear equations. Many calculators and computer algebra systems, such as Mathematica or Maple, have built-in functions for solving systems of linear equations. **Q: What if I have a system of linear equations with complex coefficients?** ------------------------------------------------------------------------- A: If you have a system of linear equations with complex coefficients, then you can use the same methods as before, but you will need to use complex numbers and complex arithmetic. You can also use matrix methods, such as Gaussian elimination or LU decomposition, to solve the system. **Q: Can I use a system of linear equations to model real-world problems?** ------------------------------------------------------------------------- A: Yes, you can use a system of linear equations to model real-world problems. Many real-world problems, such as economics, physics, and engineering, can be modeled using systems of linear equations. **Q: What are some common applications of systems of linear equations?** ------------------------------------------------------------------------- A: Some common applications of systems of linear equations include: * Economics: modeling supply and demand curves * Physics: modeling motion and forces * Engineering: designing electrical circuits and mechanical systems * Computer science: modeling algorithms and data structures **Q: Can I use a system of linear equations to solve a system of nonlinear equations?** ------------------------------------------------------------------------- A: No, you cannot use a system of linear equations to solve a system of nonlinear equations. Nonlinear equations are equations that are not linear, meaning that they do not have a constant slope. To solve a system of nonlinear equations, you will need to use more advanced methods, such as numerical methods or algebraic methods. **Q: What are some common mistakes to avoid when solving systems of linear equations?** ------------------------------------------------------------------------- A: Some common mistakes to avoid when solving systems of linear equations include: * Not checking for consistency before solving the system * Not using the correct method for the specific system * Not checking for solutions that are not unique * Not using a calculator or computer to check the solution **Q: Can I use a system of linear equations to solve a system of inequalities?** ------------------------------------------------------------------------- A: No, you cannot use a system of linear equations to solve a system of inequalities. Inequalities are statements that compare two quantities, and they do not have a solution in the same way that equations do. To solve a system of inequalities, you will need to use more advanced methods, such as graphical methods or algebraic methods. **Conclusion** ---------- In this article, we have answered some common questions about solving systems of linear equations. We have discussed the two main methods for solving systems of linear equations, the substitution method and the elimination method, and we have provided some tips for choosing between the two methods. We have also discussed some common applications of systems of linear equations and some common mistakes to avoid when solving them.