Solve The System Of Equations Given Below.$ \begin{aligned} 8x + 4y &= 16 \\ 7y &= 15 - X \end{aligned} $A. (2, 1) B. (-2, 4) C. (1, 2) D. (4, -2)

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 given system of equations to demonstrate the steps involved in solving such a system.

The System of Equations

The given system of equations is:

$$\begin{aligned} 8x + 4y &= 16 \\ 7y &= 15 - x \end{aligned}$$

Step 1: Isolate One Variable

To solve the system of equations, we can start by isolating one variable in one of the equations. Let's isolate the variable $y$ in the second equation.

$$7y = 15 - x$$

We can divide both sides of the equation by 7 to get:

$$y = \frac{15 - x}{7}$$

Step 2: Substitute the Expression into the First Equation

Now that we have an expression for $y$, we can substitute it into the first equation to get:

$$8x + 4\left(\frac{15 - x}{7}\right) = 16$$

Step 3: Simplify the Equation

To simplify the equation, we can multiply both sides by 7 to get rid of the fraction:

$$56x + 4(15 - x) = 112$$

Expanding the equation, we get:

$$56x + 60 - 4x = 112$$

Combine like terms:

$$52x + 60 = 112$$

Subtract 60 from both sides:

$$52x = 52$$

Divide both sides by 52:

$$x = 1$$

Step 4: Find the Value of the Other Variable

Now that we have the value of $x$, we can substitute it into one of the original equations to find the value of the other variable. Let's use the second equation:

$$7y = 15 - x$$

Substitute $x = 1$:

$$7y = 15 - 1$$

Simplify:

$$7y = 14$$

Divide both sides by 7:

$$y = 2$$

Conclusion

We have solved the system of linear equations and found the values of the variables $x$ and $y$. The solution is:

$$x = 1$$

$$y = 2$$

Therefore, the correct answer is:

C. (1, 2)

Tips and Tricks

• When solving a system of linear equations, it's often helpful to isolate one variable in one of the equations.
• Use substitution or elimination methods to solve the system.
• Check your solution by plugging the values back into the original equations.

Practice Problems

Try solving the following system of linear equations:

$$\begin{aligned} 2x + 3y &= 12 \\ x - 2y &= -3 \end{aligned}$$

Introduction

In our previous article, we solved a system of linear equations using the substitution method. In this article, we will answer some common questions related to 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: What are the different 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 isolating one variable in one of the equations and substituting it into the other equation.
2. Elimination Method: This method involves adding or subtracting the equations to eliminate one of the variables.

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 already in a form where one variable is isolated, the substitution method may be easier to use. If the equations are in a form where the variables are added or subtracted, the elimination method may be easier to use.

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

In this case, you can use the same methods as before, but you may need to use additional steps to isolate the variables. One way to do this is to use the substitution method to solve for one variable, and then use the elimination method to solve for the remaining variables.

Q: How do I check my solution?

To check your solution, plug the values back into the original equations and make sure they are true. If the values satisfy both equations, then you have found the correct solution.

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

In this case, the equations are inconsistent, and there is no solution. This can happen if the equations are contradictory, such as:

$$\begin{aligned} x + y &= 2 \\ x + y &= 3 \end{aligned}$$

In this case, there is no value of $x$ and $y$ that can satisfy both equations.

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

In this case, the equations are dependent, and there are infinitely many solutions. This can happen if the equations are equivalent, such as:

$$\begin{aligned} x + y &= 2 \\ 2x + 2y &= 4 \end{aligned}$$

In this case, any value of $x$ and $y$ that satisfies the first equation will also satisfy the second equation.

Conclusion

Solving a system of linear equations can be a challenging task, but with the right methods and techniques, it can be done. Remember to choose the right method for the problem, check your solution, and be aware of the different types of solutions that can occur.

Tips and Tricks

• Always check your solution by plugging the values back into the original equations.
• Use the substitution method when one variable is isolated, and the elimination method when the variables are added or subtracted.
• Be aware of the different types of solutions that can occur, including no solution, one solution, and infinitely many solutions.

Practice Problems

Try solving the following system of linear equations:

$$\begin{aligned} x + 2y &= 6 \\ 3x - 2y &= 2 \end{aligned}$$

Use the substitution method to solve the system and find the values of the variables.