Solve The Following System Of Equations. Enter The $y$-coordinate Of The Solution. Round Your Answer To The Nearest Tenth.$\begin{array}{l} 5x + 2y = 7 \\ -2x + 6y = 9 \end{array}$Answer Here: ___________

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 focus on solving a system of two linear equations with two variables.

The System of Equations

The given system of equations is:

$$\begin{array}{l} 5x + 2y = 7 \\ -2x + 6y = 9 \end{array}$$

Our goal is to find the values of $x$ and $y$ that satisfy both equations.

Method 1: Substitution Method

One way to solve this system of equations is to use the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Let's solve the first equation for $x$:

$$5x + 2y = 7$$

Subtracting $2y$ from both sides gives:

$$5x = 7 - 2y$$

Dividing both sides by 5 gives:

$$x = \frac{7 - 2y}{5}$$

Now, substitute this expression for $x$ into the second equation:

$$-2x + 6y = 9$$

Substituting $x = \frac{7 - 2y}{5}$ gives:

$$-2\left(\frac{7 - 2y}{5}\right) + 6y = 9$$

Simplifying this equation gives:

$$-\frac{14}{5} + \frac{4y}{5} + 6y = 9$$

Multiplying both sides by 5 gives:

$$-14 + 4y + 30y = 45$$

Combine like terms:

$$34y = 59$$

Dividing both sides by 34 gives:

$$y = \frac{59}{34}$$

Now that we have found the value of $y$, we can substitute it back into one of the original equations to find the value of $x$. Let's use the first equation:

$$5x + 2y = 7$$

Substituting $y = \frac{59}{34}$ gives:

$$5x + 2\left(\frac{59}{34}\right) = 7$$

Simplifying this equation gives:

$$5x + \frac{59}{17} = 7$$

Multiplying both sides by 17 gives:

$$85x + 59 = 119$$

Subtracting 59 from both sides gives:

$$85x = 60$$

Dividing both sides by 85 gives:

$$x = \frac{60}{85}$$

Simplifying this fraction gives:

$$x = \frac{12}{17}$$

Method 2: Elimination Method

Another way to solve this system of equations is to use the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables.

Let's add the two equations together:

$$5x + 2y = 7$$

$$-2x + 6y = 9$$

Adding the two equations gives:

$$3x + 8y = 16$$

Now, we can solve this new equation for $x$:

$$3x + 8y = 16$$

Subtracting $8y$ from both sides gives:

$$3x = 16 - 8y$$

Dividing both sides by 3 gives:

$$x = \frac{16 - 8y}{3}$$

Now, substitute this expression for $x$ into one of the original equations. Let's use the first equation:

$$5x + 2y = 7$$

Substituting $x = \frac{16 - 8y}{3}$ gives:

$$5\left(\frac{16 - 8y}{3}\right) + 2y = 7$$

Simplifying this equation gives:

$$\frac{80 - 40y}{3} + 2y = 7$$

Multiplying both sides by 3 gives:

$$80 - 40y + 6y = 21$$

Combine like terms:

$$-34y = -59$$

Dividing both sides by -34 gives:

$$y = \frac{59}{34}$$

Now that we have found the value of $y$, we can substitute it back into one of the original equations to find the value of $x$. Let's use the first equation:

$$5x + 2y = 7$$

Substituting $y = \frac{59}{34}$ gives:

$$5x + 2\left(\frac{59}{34}\right) = 7$$

Simplifying this equation gives:

$$5x + \frac{59}{17} = 7$$

Multiplying both sides by 17 gives:

$$85x + 59 = 119$$

Subtracting 59 from both sides gives:

$$85x = 60$$

Dividing both sides by 85 gives:

$$x = \frac{60}{85}$$

Simplifying this fraction gives:

$$x = \frac{12}{17}$$

Conclusion

In this article, we have solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. Both methods have led to the same solution: $x = \frac{12}{17}$ and $y = \frac{59}{34}$. We have also rounded the value of $y$ to the nearest tenth, as required by the problem.

Answer

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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 involve the same set of variables. In other words, it is a collection of equations that are all related to each other through the variables.

Q: How do I know which method to use to solve a system of linear equations?

A: There are two main methods to solve a system of linear equations: the substitution method and the elimination method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables. You can choose the method that you prefer or that seems easier to you.

Q: What is the substitution method?

A: The substitution method 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 already solved for one variable.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of one of the variables are the same in both equations.

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 with each other. In other words, if the equations are not contradictory, then there is a solution.

Q: What if the system of linear equations has no solution?

A: If the system of linear equations has no solution, it means that the equations are contradictory. In other words, the equations are saying two different things at the same time, and it is not possible to satisfy both equations.

Q: What if the system of linear equations has infinitely many solutions?

A: If the system of linear equations has infinitely many solutions, it means that the equations are dependent on each other. In other words, one equation is a multiple of the other equation, and there are many possible solutions.

Q: How do I know if a system of linear equations has infinitely many solutions?

A: A system of linear equations has infinitely many solutions if the two equations are dependent on each other. In other words, if one equation is a multiple of the other equation, then there are many possible solutions.

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

A: Yes, you can use a graphing calculator to solve a system of linear equations. Graphing calculators can help you visualize the equations and find the solution.

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

A: Yes, you can use a computer program to solve a system of linear equations. Computer programs can help you solve the equations quickly and accurately.

Conclusion

In this article, we have answered some common questions about solving a system of linear equations. We have discussed the substitution method and the elimination method, and we have talked about how to know if a system of linear equations has a solution, no solution, or infinitely many solutions. We have also discussed how to use a graphing calculator and a computer program to solve a system of linear equations.