Find The Solution Of The System Of Equations.$\[ \begin{array}{l} 3x - 8y = 19 \\ 3x + 5y = -46 \end{array} \\]( \[$ \square \$\] , \[$ \square \$\] )
Introduction
Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple equations simultaneously. 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.
What is a System of Linear Equations?
A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. The system of equations is said to be consistent if it has a solution, and inconsistent if it does not have a solution.
The Method of Substitution
The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one of the equations is easily solvable for one variable.
Step 1: Solve One Equation for One Variable
Let's consider the following system of equations:
{ \begin{array}{l} 3x - 8y = 19 \\ 3x + 5y = -46 \end{array} \}
We can solve the first equation for x:
3x = 19 + 8y
x = (19 + 8y) / 3
Step 2: Substitute the Expression into the Other Equation
Now, we can substitute the expression for x into the second equation:
3((19 + 8y) / 3) + 5y = -46
19 + 8y + 5y = -46
13y = -65
y = -65 / 13
y = -5
Step 3: Find the Value of the Other Variable
Now that we have the value of y, we can find the value of x by substituting y into one of the original equations:
3x - 8(-5) = 19
3x + 40 = 19
3x = -21
x = -21 / 3
x = -7
The Method of Elimination
The method of elimination involves adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of one variable are the same in both equations.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to make the coefficients of that variable the same in both equations. We can do this by multiplying the equations by necessary multiples:
Equation 1: 3x - 8y = 19
Equation 2: 3x + 5y = -46
We can multiply the first equation by 1 and the second equation by 1.
Step 2: Add or Subtract the Equations
Now, we can add or subtract the equations to eliminate one of the variables. Let's add the equations:
(3x - 8y) + (3x + 5y) = 19 + (-46)
6x - 3y = -27
Step 3: Solve for the Variable
Now that we have the equation with one variable eliminated, we can solve for that variable:
6x = -27 + 3y
x = (-27 + 3y) / 6
x = (-9 + y) / 2
Step 4: Find the Value of the Other Variable
Now that we have the value of x, we can find the value of y by substituting x into one of the original equations:
3x - 8y = 19
3((-9 + y) / 2) - 8y = 19
-27 + 3y - 16y = 38
-13y = 65
y = -5
Step 5: Find the Value of the Other Variable
Now that we have the value of y, we can find the value of x by substituting y into one of the original equations:
3x + 5y = -46
3x + 5(-5) = -46
3x - 25 = -46
3x = -21
x = -7
Conclusion
In this article, we have discussed the method of substitution and elimination for solving a system of linear equations. We have used the method of substitution to solve a system of two linear equations with two variables. We have also used the method of elimination to solve a system of two linear equations with two variables. We have shown that both methods can be used to find the solution of a system of linear equations.
Final Answer
The final answer is .
Introduction
Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple equations simultaneously. 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.
Q&A: Solving a System 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 two or more variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.
Q: What are the methods of solving a system of linear equations?
A: There are two main methods of solving a system of linear equations: the method of substitution and the method of elimination.
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: What is the method of elimination?
A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables.
Q: How do I choose which method to use?
A: You can choose which method to use based on the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, you can use the method of elimination. If the coefficients of one variable are different in both equations, you can use the method of substitution.
Q: What if I have a system of three or more linear equations?
A: If you have a system of three or more linear equations, you can use the method of substitution or elimination to solve for two variables, and then use the third equation to solve for the third variable.
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. This can happen if the equations are contradictory, such as 2x + 3y = 5 and 2x + 3y = 10.
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. This can happen if the equations are identical, such as 2x + 3y = 5 and 2x + 3y = 5.
Q: How do I know if a system of linear equations has a solution?
A: You can determine if a system of linear equations has a solution by checking if the equations are consistent. If the equations are consistent, the system has a solution. If the equations are inconsistent, the system has no solution.
Q: How do I know if a system of linear equations has infinitely many solutions?
A: You can determine if a system of linear equations has infinitely many solutions by checking if the equations are dependent. If the equations are dependent, the system has infinitely many solutions.
Conclusion
In this article, we have discussed the method of substitution and elimination for solving a system of linear equations. We have also answered some common questions about solving a system of linear equations. We hope that this article has been helpful in understanding how to solve a system of linear equations.
Final Answer
The final answer is .