Solve The System Of Equations:${ \begin{array}{l} 2x - 3y = 5 \ 4x + 5y = 53 \end{array} }$
Introduction to Systems of Linear Equations
Systems of linear equations are a fundamental concept in mathematics, particularly in algebra and geometry. They consist of two or more linear equations that involve the same variables. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given system of equations as an example to demonstrate the steps involved in solving it.
The Given System of Equations
The system of equations we will be solving is:
{ \begin{array}{l} 2x - 3y = 5 \\ 4x + 5y = 53 \end{array} \}
Method 1: Substitution Method
One way to solve this system of equations is by using the substitution method. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.
Step 1: Solve the First Equation for x
We can solve the first equation for x by adding 3y to both sides of the equation and then dividing both sides by 2.
{ 2x - 3y = 5 \}
{ 2x = 5 + 3y \}
{ x = \frac{5 + 3y}{2} \}
Step 2: Substitute the Expression for x into the Second Equation
Now that we have an expression for x, we can substitute it into the second equation.
{ 4x + 5y = 53 \}
{ 4\left(\frac{5 + 3y}{2}\right) + 5y = 53 \}
{ 2(5 + 3y) + 5y = 53 \}
{ 10 + 6y + 5y = 53 \}
{ 11y = 43 \}
{ y = \frac{43}{11} \}
Step 3: Find the Value of x
Now that we have the value of y, we can substitute it back into the expression for x that we found in Step 1.
{ x = \frac{5 + 3y}{2} \}
{ x = \frac{5 + 3\left(\frac{43}{11}\right)}{2} \}
{ x = \frac{5 + \frac{129}{11}}{2} \}
{ x = \frac{\frac{55}{11} + \frac{129}{11}}{2} \}
{ x = \frac{\frac{184}{11}}{2} \}
{ x = \frac{184}{22} \}
{ x = \frac{92}{11} \}
Method 2: Elimination Method
Another way to solve this system of equations is by using the elimination method. This method involves adding or subtracting the two equations to eliminate one of the variables.
Step 1: Multiply the Two Equations by Necessary Multiples
To eliminate one of the variables, we need to make the coefficients of either x or y the same in both equations. We can do this by multiplying the two equations by necessary multiples.
{ 2x - 3y = 5 \}
{ 4x + 5y = 53 \}
{ (2x - 3y) \times 5 = (5) \times 5 \}
{ 10x - 15y = 25 \}
{ (4x + 5y) \times 3 = (53) \times 3 \}
{ 12x + 15y = 159 \}
Step 2: Add the Two Equations
Now that we have the two equations with the same coefficients for y, we can add them to eliminate y.
{ (10x - 15y) + (12x + 15y) = 25 + 159 \}
{ 22x = 184 \}
{ x = \frac{184}{22} \}
{ x = \frac{92}{11} \}
Step 3: Find the Value of y
Now that we have the value of x, we can substitute it back into one of the original equations to find the value of y.
{ 2x - 3y = 5 \}
{ 2\left(\frac{92}{11}\right) - 3y = 5 \}
{ \frac{184}{11} - 3y = 5 \}
{ -3y = 5 - \frac{184}{11} \}
{ -3y = \frac{55 - 184}{11} \}
{ -3y = \frac{-129}{11} \}
{ y = \frac{129}{33} \}
{ y = \frac{43}{11} \}
Conclusion
In this article, we have solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. We have found the values of x and y that satisfy both equations. The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the two equations to eliminate one of the variables. Both methods are useful tools for solving systems of linear equations, and the choice of method depends on the specific problem and the preferences of the solver.
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 variables. In this article, we have focused on solving a system of two linear equations with two variables.
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 of the equations for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the two 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 problem and the preferences of the solver. If one of the equations is already solved for one variable, the substitution method may be easier to use. If the coefficients of the variables are the same in both equations, the elimination method may be easier to use.
Q: What if I have a system of linear equations with more than two variables?
A: If you have a system of linear equations with more than two variables, you can use the same methods as before, but you may need to use additional techniques, such as substitution or elimination, to solve for each variable.
Q: Can I use a calculator to solve systems of linear equations?
A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations, and you can also use software programs, such as graphing calculators or computer algebra systems, to solve systems of linear equations.
Q: What if I have a system of linear equations with no solution?
A: If you have a system of linear equations with no solution, it means that the equations are inconsistent, and there is no value of the variables that satisfies both equations. This can happen if the equations are contradictory, or if one of the equations is a linear combination of the other equation.
Q: What if I have a system of linear equations with infinitely many solutions?
A: If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent, and there are many values of the variables that satisfy both equations. This can happen if one of the equations is a linear combination of the other equation.
Q: Can I use systems of linear equations to model real-world problems?
A: Yes, you can use systems of linear equations to model real-world problems. Systems of linear equations can be used to model a wide range of problems, including economics, physics, engineering, and more.
Q: What are some common applications of systems of linear equations?
A: Some common applications of systems of linear equations include:
- Modeling the cost of producing a product
- Determining the amount of money in a bank account
- Finding the dimensions of a rectangle
- Solving problems in physics and engineering
- Modeling population growth and decline
Q: Can I use systems of linear equations to solve problems in other areas of mathematics?
A: Yes, you can use systems of linear equations to solve problems in other areas of mathematics, including algebra, geometry, and calculus.
Q: What are some tips for solving systems of linear equations?
A: Some tips for solving systems of linear equations include:
- Read the problem carefully and understand what is being asked
- Choose the method that is easiest to use
- Check your work to make sure that the solution is correct
- Use a calculator or computer algebra system to check your work
- Practice solving systems of linear equations to become more confident and proficient.