Solve The System Of Equations:$\[ \begin{array}{l} x - 7y = 4 \\ 3x + Y = -10 \end{array} \\]
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Introduction
In mathematics, a system of linear equations is a set of two or more linear 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. We will use the method of substitution and elimination to find the solution.
The System of Equations
The system of equations we will be solving is:
Method of Substitution
One way to solve this system of equations is by using the method of substitution. 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 7y to both sides of the equation:
Step 2: Substitute the Expression for x into the Second Equation
Now, we can substitute the expression for x into the second equation:
Step 3: Simplify the Equation
We can simplify the equation by distributing the 3 to the terms inside the parentheses:
Step 4: Combine Like Terms
We can combine the like terms on the left-hand side of the equation:
Step 5: Subtract 12 from Both Sides
We can subtract 12 from both sides of the equation to isolate the term with the variable:
Step 6: Divide Both Sides by 22
We can divide both sides of the equation by 22 to solve for y:
Step 7: Substitute the Value of y into One of the Original Equations
Now that we have the value of y, we can substitute it into one of the original equations to solve for x. We will use the first equation:
Step 8: Simplify the Equation
We can simplify the equation by multiplying -7 and -1:
Step 9: Subtract 7 from Both Sides
We can subtract 7 from both sides of the equation to isolate the term with the variable:
Method of Elimination
Another way to solve this system of equations is by using the method of elimination. This method involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same.
Step 1: Multiply the First Equation by 1 and the Second Equation by 7
We can multiply the first equation by 1 and the second equation by 7:
Step 2: Add Both Equations
We can add both equations to eliminate the variable y:
Step 3: Simplify the Equation
We can simplify the equation by combining like terms:
Step 4: Divide Both Sides by 22
We can divide both sides of the equation by 22 to solve for x:
Step 5: Substitute the Value of x into One of the Original Equations
Now that we have the value of x, we can substitute it into one of the original equations to solve for y. We will use the first equation:
Step 6: Simplify the Equation
We can simplify the equation by adding 7y to both sides:
Step 7: Divide Both Sides by -7
We can divide both sides of the equation by -7 to solve for y:
Conclusion
In this article, we have solved a system of two linear equations with two variables using the method of substitution and elimination. We have shown that both methods can be used to find the solution to the system of equations. The solution to the system of equations is x = -3 and y = -1.
Final Answer
The final answer is x = -3 and y = -1.
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Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the 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 method of substitution and the method of elimination.
Q: What is the method of substitution?
A: The method of substitution involves solving one of the equations 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 multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same, and then adding both equations to eliminate the variable y.
Q: How do I know which method to use?
A: You can use either method, but the method of elimination is often easier to use when the coefficients of the variables are large.
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, or you can use a matrix method.
Q: What is a matrix method?
A: A matrix method involves representing the system of equations as a matrix and then using row operations to solve the system.
Q: How do I know if a system of linear equations has a solution?
A: A system of linear equations has a solution if the two equations are consistent, meaning that they have the same solution.
Q: What if a system of linear equations has no solution?
A: If a system of linear equations has no solution, it means that the two equations are inconsistent, meaning that they have no common solution.
Q: What if a system of linear equations has infinitely many solutions?
A: If a system of linear equations has infinitely many solutions, it means that the two equations are dependent, meaning that one equation is a multiple of the other.
Q: How do I graph a system of linear equations?
A: To graph a system of linear equations, you can plot the two equations on a coordinate plane and then find the point of intersection, which is the solution to the system.
Q: What are some real-world applications of solving systems of linear equations?
A: Solving systems of linear equations has many real-world applications, including physics, engineering, economics, and computer science.
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.
Q: How do I check my work when solving a system of linear equations?
A: To check your work when solving a system of linear equations, you can plug the solution back into both equations to make sure that it satisfies both equations.
Q: What if I make a mistake when solving a system of linear equations?
A: If you make a mistake when solving a system of linear equations, you can go back and recheck your work to find the error.
Q: Can I use technology to help me solve systems of linear equations?
A: Yes, you can use technology, such as graphing calculators or computer software, to help you solve systems of linear equations.
Q: How do I know if a system of linear equations is consistent or inconsistent?
A: A system of linear equations is consistent if it has a solution, and it is inconsistent if it has no solution.
Q: What is the difference between a consistent and an inconsistent system of linear equations?
A: A consistent system of linear equations has a solution, while an inconsistent system of linear equations has no solution.
Q: Can a system of linear equations have both a consistent and an inconsistent solution?
A: No, a system of linear equations cannot have both a consistent and an inconsistent solution.
Q: How do I determine if a system of linear equations is dependent or independent?
A: A system of linear equations is dependent if one equation is a multiple of the other, and it is independent if the two equations are not multiples of each other.
Q: What is the difference between a dependent and an independent system of linear equations?
A: A dependent system of linear equations has infinitely many solutions, while an independent system of linear equations has a unique solution.
Q: Can a system of linear equations be both dependent and independent?
A: No, a system of linear equations cannot be both dependent and independent.
Q: How do I know if a system of linear equations is homogeneous or nonhomogeneous?
A: A system of linear equations is homogeneous if all the constants are zero, and it is nonhomogeneous if not all the constants are zero.
Q: What is the difference between a homogeneous and a nonhomogeneous system of linear equations?
A: A homogeneous system of linear equations has only the trivial solution, while a nonhomogeneous system of linear equations has a nontrivial solution.
Q: Can a system of linear equations be both homogeneous and nonhomogeneous?
A: No, a system of linear equations cannot be both homogeneous and nonhomogeneous.
Q: How do I determine if a system of linear equations is solvable or unsolvable?
A: A system of linear equations is solvable if it has a solution, and it is unsolvable if it has no solution.
Q: What is the difference between a solvable and an unsolvable system of linear equations?
A: A solvable system of linear equations has a solution, while an unsolvable system of linear equations has no solution.
Q: Can a system of linear equations be both solvable and unsolvable?
A: No, a system of linear equations cannot be both solvable and unsolvable.
Q: How do I know if a system of linear equations is consistent or inconsistent?
A: A system of linear equations is consistent if it has a solution, and it is inconsistent if it has no solution.
Q: What is the difference between a consistent and an inconsistent system of linear equations?
A: A consistent system of linear equations has a solution, while an inconsistent system of linear equations has no solution.
Q: Can a system of linear equations have both a consistent and an inconsistent solution?
A: No, a system of linear equations cannot have both a consistent and an inconsistent solution.
Q: How do I determine if a system of linear equations is dependent or independent?
A: A system of linear equations is dependent if one equation is a multiple of the other, and it is independent if the two equations are not multiples of each other.
Q: What is the difference between a dependent and an independent system of linear equations?
A: A dependent system of linear equations has infinitely many solutions, while an independent system of linear equations has a unique solution.
Q: Can a system of linear equations be both dependent and independent?
A: No, a system of linear equations cannot be both dependent and independent.
Q: How do I know if a system of linear equations is homogeneous or nonhomogeneous?
A: A system of linear equations is homogeneous if all the constants are zero, and it is nonhomogeneous if not all the constants are zero.
Q: What is the difference between a homogeneous and a nonhomogeneous system of linear equations?
A: A homogeneous system of linear equations has only the trivial solution, while a nonhomogeneous system of linear equations has a nontrivial solution.
Q: Can a system of linear equations be both homogeneous and nonhomogeneous?
A: No, a system of linear equations cannot be both homogeneous and nonhomogeneous.
Q: How do I determine if a system of linear equations is solvable or unsolvable?
A: A system of linear equations is solvable if it has a solution, and it is unsolvable if it has no solution.
Q: What is the difference between a solvable and an unsolvable system of linear equations?
A: A solvable system of linear equations has a solution, while an unsolvable system of linear equations has no solution.
Q: Can a system of linear equations be both solvable and unsolvable?
A: No, a system of linear equations cannot be both solvable and unsolvable.
Q: How do I know if a system of linear equations is consistent or inconsistent?
A: A system of linear equations is consistent if it has a solution, and it is inconsistent if it has no solution.
Q: What is the difference between a consistent and an inconsistent system of linear equations?
A: A consistent system of linear equations has a solution, while an inconsistent system of linear equations has no solution.
Q: Can a system of linear equations have both a consistent and an inconsistent solution?
A: No, a system of linear equations cannot have both a consistent and an inconsistent solution.