Find The Solution Of The System Of Equations:$\[ \begin{align*} -3x + 10y &= 17 \\ 3x + 7y &= 17 \end{align*} \\]

Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and linear algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. 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 as an example and provide a step-by-step solution.

The System of Equations

The given system of equations is:

{ \begin{align*} -3x + 10y &= 17 \\ 3x + 7y &= 17 \end{align*} \}

This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations simultaneously.

Method 1: Substitution Method

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

Step 1: Solve the First Equation for x

We can start by solving the first equation for x:

{ -3x + 10y = 17 \}

Subtracting 10y from both sides gives:

{ -3x = 17 - 10y \}

Dividing both sides by -3 gives:

{ x = \frac{17 - 10y}{-3} \}

Simplifying the expression gives:

{ x = \frac{10y - 17}{3} \}

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

Now that we have an expression for x, we can substitute it into the second equation:

{ 3x + 7y = 17 \}

Substituting x = (10y - 17)/3 gives:

{ 3\left(\frac{10y - 17}{3}\right) + 7y = 17 \}

Simplifying the expression gives:

{ 10y - 17 + 7y = 17 \}

Combine like terms:

{ 17y - 17 = 17 \}

Add 17 to both sides:

{ 17y = 34 \}

Divide both sides by 17:

{ y = 2 \}

Step 3: Find the Value of x

Now that we have the value of y, we can find the value of x by substituting y = 2 into the expression for x:

{ x = \frac{10y - 17}{3} \}

Substituting y = 2 gives:

{ x = \frac{10(2) - 17}{3} \}

Simplifying the expression gives:

{ x = \frac{20 - 17}{3} \}

Simplifying further gives:

{ x = \frac{3}{3} \}

Simplifying even further gives:

{ x = 1 \}

Method 2: Elimination Method

Another way to solve this system is by using the elimination method. This method involves adding or subtracting the equations to eliminate one variable.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one variable, we need to multiply the equations by necessary multiples such that the coefficients of x or y in both equations are the same.

Multiplying the first equation by 3 gives:

{ -9x + 30y = 51 \}

Multiplying the second equation by 1 gives:

{ 3x + 7y = 17 \}

Step 2: Add or Subtract the Equations

Now that we have the equations multiplied by necessary multiples, we can add or subtract them to eliminate one variable.

Adding the equations gives:

{ -9x + 30y + 3x + 7y = 51 + 17 \}

Combine like terms:

{ -6x + 37y = 68 \}

Subtracting the second equation from the first equation gives:

{ -9x + 30y - (3x + 7y) = 51 - 17 \}

Simplifying the expression gives:

{ -12x + 23y = 34 \}

Step 3: Solve for the Variable

Now that we have the equation with one variable eliminated, we can solve for that variable.

We can solve for y by using the equation -6x + 37y = 68.

However, we notice that the equation -6x + 37y = 68 is not the same as the equation 17y = 34 that we obtained using the substitution method.

This is because the elimination method involves adding or subtracting the equations, which can result in a different equation.

To solve for y, we can use the equation 17y = 34.

Dividing both sides by 17 gives:

{ y = 2 \}

Step 4: Find the Value of x

Now that we have the value of y, we can find the value of x by substituting y = 2 into one of the original equations.

Substituting y = 2 into the first equation gives:

{ -3x + 10(2) = 17 \}

Simplifying the expression gives:

{ -3x + 20 = 17 \}

Subtracting 20 from both sides gives:

{ -3x = -3 \}

Dividing both sides by -3 gives:

{ x = 1 \}

Conclusion

In this article, we solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. We found that both methods resulted in the same solution: x = 1 and y = 2.

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 variable.

Both methods are useful for solving systems of linear equations, and the choice of method depends on the specific problem and the preference of the solver.

Final Answer

The final answer is: $\boxed{1, 2}$

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and linear algebra. In our previous article, we provided a step-by-step guide on how to solve a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. In this article, we will answer some frequently asked questions (FAQs) about solving systems of linear equations.

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 two or more variables. Each equation in the system is a linear equation, which means that it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

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

A: To determine if a system of linear equations has a solution, you can use the following methods:

• Check if the equations are consistent. If the equations are consistent, then the system has a solution.
• Check if the equations are inconsistent. If the equations are inconsistent, then the system does not have a solution.
• Use the method of substitution or elimination to solve the system. If the system has a solution, then the method will give you the values of the variables.

Q: What is the difference between the substitution method and the elimination method?

A: 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 variable.

Q: Which method is better, the substitution method or the elimination method?

A: Both methods are useful for solving systems of linear equations, and the choice of method depends on the specific problem and the preference of the solver. The substitution method is often easier to use when one of the variables is already isolated in one of the equations. The elimination method is often easier to use when the coefficients of the variables in both equations are the same.

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

A: To determine if a system of linear equations has a unique solution, you can use the following methods:

• Check if the equations are consistent and have a unique solution. If the equations are consistent and have a unique solution, then the system has a unique solution.
• Check if the equations are inconsistent. If the equations are inconsistent, then the system does not have a unique solution.
• Use the method of substitution or elimination to solve the system. If the system has a unique solution, then the method will give you the values of the variables.

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

A: A system of linear equations is a set of linear equations that involve two or more variables. A system of nonlinear equations is a set of nonlinear equations that involve two or more variables. Nonlinear equations are equations that cannot be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I solve a system of nonlinear equations?

A: Solving a system of nonlinear equations is more complex than solving a system of linear equations. There are several methods that can be used to solve a system of nonlinear equations, including:

• The method of substitution
• The method of elimination
• The method of numerical methods
• The method of graphical methods

Conclusion

In this article, we answered some frequently asked questions (FAQs) about solving systems of linear equations. We provided information on how to determine if a system of linear equations has a solution, how to choose between the substitution method and the elimination method, and how to determine if a system of linear equations has a unique solution. We also provided information on the difference between a system of linear equations and a system of nonlinear equations, and how to solve a system of nonlinear equations.

Final Answer

The final answer is: There is no final answer, as this article is a Q&A article that provides information and answers to frequently asked questions about solving systems of linear equations.