Solve The Following System:${ \begin{align*} y &= X + 3 \ 3x + Y &= 19 \end{align*} }$A. { (7, 4)$}$B. { \left(-4, \frac{14}{4}\right)$}$C. { (4, 7)$}$D. { (4, -7)$}$

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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 involves finding the values of the variables that satisfy all the equations in the system. In this article, we will solve a system of two linear equations using the substitution method.

The System of Linear Equations

The system of linear equations we will solve is:

y=x+33x+y=19\begin{align*} y &= x + 3 \\ 3x + y &= 19 \end{align*}

Step 1: Solve the First Equation for y

The first equation is already solved for y, so we can write it as:

y=x+3y = x + 3

Step 2: Substitute the Expression for y into the Second Equation

We will substitute the expression for y from the first equation into the second equation:

3x+(x+3)=193x + (x + 3) = 19

Step 3: Simplify the Equation

We will simplify the equation by combining like terms:

3x+x+3=193x + x + 3 = 19

4x+3=194x + 3 = 19

Step 4: Subtract 3 from Both Sides

We will subtract 3 from both sides of the equation to isolate the term with the variable:

4x+3βˆ’3=19βˆ’34x + 3 - 3 = 19 - 3

4x=164x = 16

Step 5: Divide Both Sides by 4

We will divide both sides of the equation by 4 to solve for x:

4x4=164\frac{4x}{4} = \frac{16}{4}

x=4x = 4

Step 6: Find the Value of y

Now that we have the value of x, we can substitute it into the first equation to find the value of y:

y=x+3y = x + 3

y=4+3y = 4 + 3

y=7y = 7

Conclusion

The solution to the system of linear equations is x = 4 and y = 7. Therefore, the correct answer is:

(C) (4, 7)

Why This Method Works

The substitution method works because we are able to substitute the expression for y from the first equation into the second equation. This allows us to eliminate the variable y and solve for the variable x. Once we have the value of x, we can substitute it into the first equation to find the value of y.

Advantages of the Substitution Method

The substitution method has several advantages. It is a simple and straightforward method that is easy to understand and apply. It also allows us to solve systems of linear equations with any number of variables.

Disadvantages of the Substitution Method

The substitution method has several disadvantages. It can be time-consuming and labor-intensive, especially for large systems of linear equations. It also requires us to have a clear understanding of the equations and the variables involved.

Real-World Applications

The substitution method has many real-world applications. It is used in a variety of fields, including physics, engineering, and economics. It is also used in computer science and data analysis.

Conclusion

In conclusion, the substitution method is a powerful tool for solving systems of linear equations. It is a simple and straightforward method that is easy to understand and apply. It has many real-world applications and is used in a variety of fields.

Final Answer

The final answer is:

Introduction

In our previous article, we solved a system of linear equations using the substitution method. In this article, we will answer some frequently asked questions about solving systems of linear equations.

Q: 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. For example:

y=x+33x+y=19\begin{align*} y &= x + 3 \\ 3x + y &= 19 \end{align*}

Q: How do I know if a system of linear equations has a solution?

A system of linear equations has a solution if and only if the two equations are consistent. In other words, if the two equations have the same solution, then the system has a solution.

Q: What is the difference between a consistent and inconsistent system of linear equations?

A consistent system of linear equations has a solution, while an inconsistent system of linear equations does not have a solution.

Q: How do I determine if a system of linear equations is consistent or inconsistent?

To determine if a system of linear equations is consistent or inconsistent, you can use the following methods:

  • Substitution method: Substitute the expression for one variable from one equation into the other equation.
  • Elimination method: Add or subtract the two equations to eliminate one variable.
  • Graphical method: Graph the two equations on a coordinate plane and see if they intersect.

Q: What is the substitution method?

The substitution method is a method for solving systems of linear equations by substituting the expression for one variable from one equation into the other equation.

Q: What is the elimination method?

The elimination method is a method for solving systems of linear equations by adding or subtracting the two equations to eliminate one variable.

Q: What is the graphical method?

The graphical method is a method for solving systems of linear equations by graphing the two equations on a coordinate plane and seeing if they intersect.

Q: How do I solve a system of linear equations using the substitution method?

To solve a system of linear equations using the substitution method, follow these steps:

  1. Solve one equation for one variable: Solve one equation for one variable.
  2. Substitute the expression into the other equation: Substitute the expression for one variable from one equation into the other equation.
  3. Solve for the other variable: Solve for the other variable.

Q: How do I solve a system of linear equations using the elimination method?

To solve a system of linear equations using the elimination method, follow these steps:

  1. Add or subtract the two equations: Add or subtract the two equations to eliminate one variable.
  2. Solve for the remaining variable: Solve for the remaining variable.

Q: How do I solve a system of linear equations using the graphical method?

To solve a system of linear equations using the graphical method, follow these steps:

  1. Graph the two equations: Graph the two equations on a coordinate plane.
  2. Find the intersection point: Find the intersection point of the two graphs.
  3. Determine the solution: Determine the solution to the system of linear equations.

Conclusion

In conclusion, solving systems of linear equations is an important topic in mathematics. There are several methods for solving systems of linear equations, including the substitution method, elimination method, and graphical method. By understanding these methods, you can solve systems of linear equations and apply them to real-world problems.

Final Answer

The final answer is:

  • Substitution method: Substitute the expression for one variable from one equation into the other equation.
  • Elimination method: Add or subtract the two equations to eliminate one variable.
  • Graphical method: Graph the two equations on a coordinate plane and see if they intersect.