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

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 Linear Equations

The given system of linear equations is:

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

Step 1: Write Down the Equations

We have two linear equations:

1. $y = x + 3$
2. $3x + y = 19$

Step 2: Solve One Equation for One Variable

We can solve the first equation for $y$:

$$y = x + 3$$

Step 3: Substitute the Expression into the Second Equation

We can substitute the expression for $y$ into the second equation:

$$3x + (x + 3) = 19$$

Step 4: Simplify the Equation

We can simplify the equation by combining like terms:

$$4x + 3 = 19$$

Step 5: Solve for the Variable

We can solve for $x$ by subtracting 3 from both sides of the equation:

$$4x = 16$$

Then, we can divide both sides of the equation by 4:

$$x = 4$$

Step 6: 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:

$$y = x + 3$$

$$y = 4 + 3$$

$$y = 7$$

Conclusion

Therefore, the solution to the system of linear equations is $(4, 7)$.

Answer

The correct answer is:

• A. $(7, 4)$ is incorrect because the value of $x$ is not 7.
• B. $(-4, 7)$ is incorrect because the value of $x$ is not -4.
• C. $(4, 7)$ is correct because the values of $x$ and $y$ are both 4 and 7, respectively.
• D. $(4, -7)$ is incorrect because the value of $y$ is not -7.

Final Answer

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Introduction

In our previous article, we solved a system of linear equations using the method of substitution and elimination. 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 are solved simultaneously to find the values of the variables.

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

A system of linear equations has a solution if the two equations are consistent, meaning that they have the same solution. If the two equations are inconsistent, meaning that they have no solution, then the system has no solution.

Q: What are the methods for solving a system of linear equations?

There are two main methods for solving a system of linear equations:

1. Substitution method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
2. Elimination method: This method involves adding or subtracting the two equations to eliminate one variable.

Q: How do I choose which method to use?

The choice of method depends on the form of the equations and the variables involved. If the equations are in the form of $y = mx + b$, then the substitution method may be easier to use. If the equations are in the form of $ax + by = c$, then the elimination method may be easier to use.

Q: What if I have a system of linear equations with three or more variables?

If you have a system of linear equations with three or more variables, then you can use the same methods as before, but you may need to use more complex techniques, such as matrix operations or graphing.

Q: Can I use a calculator to solve a system of linear equations?

Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations, such as the "solve" function.

Q: What if I have a system of linear equations with no solution?

If you have a system of linear equations with no solution, then the two equations are inconsistent, meaning that they have no solution. In this case, you can say that the system has no solution.

Q: Can I have a system of linear equations with infinitely many solutions?

Yes, you can have a system of linear equations with infinitely many solutions. This occurs when the two equations are dependent, meaning that one equation is a multiple of the other.

Conclusion

Solving a system of linear equations can be a challenging task, but with the right methods and techniques, you can find the solution. Remember to choose the method that is best for the problem, and don't be afraid to use a calculator if you need to.

Final Answer

The final answer is $\boxed{yes}$, you can solve a system of linear equations using the methods and techniques described in this article.