Solve The Given System Using The Substitution Method.${ \begin{array}{l} y = 4x - 6 \ 8x - 2y = 14 \end{array} }$A. { (14,12)$}$ B. { (12,14)$}$ C. There Are An Infinite Number Of Solutions. D. There Is No Solution.

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The substitution method is a technique used to solve a system of linear equations by substituting one equation into the other. This method is particularly useful when one of the equations is already solved for one of the variables. In this article, we will use the substitution method to solve a given system of equations.

Understanding the System of Equations

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The given system of equations is:

{ \begin{array}{l} y = 4x - 6 \\ 8x - 2y = 14 \end{array} \}

The first equation is already solved for $y$, which makes it a good candidate for the substitution method. We can substitute the expression for $y$ from the first equation into the second equation.

Substituting the Expression for $y$

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

$8x - 2(4x - 6) = 14$

Simplifying the Equation

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We will simplify the equation by distributing the $-2$ to the terms inside the parentheses:

$8x - 8x + 12 = 14$

Combining Like Terms

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We will combine like terms to simplify the equation further:

$12 = 14$

Analyzing the Result

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The equation $12 = 14$ is a contradiction, which means that there is no solution to the system of equations. This is because the two sides of the equation are not equal, and therefore, there is no value of $x$ that can satisfy the equation.

Conclusion

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In conclusion, the substitution method is a useful technique for solving systems of linear equations. However, it is not always possible to find a solution to a system of equations. In this case, the system of equations has no solution, which means that there is no value of $x$ that can satisfy both equations.

Frequently Asked Questions

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Q: What is the substitution method?

A: The substitution method is a technique used to solve a system of linear equations by substituting one equation into the other.

Q: When is the substitution method useful?

A: The substitution method is particularly useful when one of the equations is already solved for one of the variables.

Q: What happens if the substitution method results in a contradiction?

A: If the substitution method results in a contradiction, it means that there is no solution to the system of equations.

Example Problems

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Problem 1

Solve the system of equations using the substitution method:

{ \begin{array}{l} y = 2x + 3 \\ x + 2y = 7 \end{array} \}

Solution

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

$x + 2(2x + 3) = 7$

We will simplify the equation by distributing the $2$ to the terms inside the parentheses:

$x + 4x + 6 = 7$

We will combine like terms to simplify the equation further:

$5x + 6 = 7$

We will subtract $6$ from both sides of the equation:

$5x = 1$

We will divide both sides of the equation by $5$:

$x = \frac{1}{5}$

We will substitute the value of $x$ into one of the original equations to find the value of $y$:

$y = 2\left(\frac{1}{5}\right) + 3$

We will simplify the equation to find the value of $y$:

$y = \frac{2}{5} + 3$

$y = \frac{2}{5} + \frac{15}{5}$

$y = \frac{17}{5}$

Therefore, the solution to the system of equations is $\left(\frac{1}{5}, \frac{17}{5}\right)$.

Problem 2

Solve the system of equations using the substitution method:

{ \begin{array}{l} y = 3x - 2 \\ 2x + 3y = 11 \end{array} \}

Solution

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

$2x + 3(3x - 2) = 11$

We will simplify the equation by distributing the $3$ to the terms inside the parentheses:

$2x + 9x - 6 = 11$

We will combine like terms to simplify the equation further:

$11x - 6 = 11$

We will add $6$ to both sides of the equation:

$11x = 17$

We will divide both sides of the equation by $11$:

$x = \frac{17}{11}$

We will substitute the value of $x$ into one of the original equations to find the value of $y$:

$y = 3\left(\frac{17}{11}\right) - 2$

We will simplify the equation to find the value of $y$:

$y = \frac{51}{11} - \frac{22}{11}$

$y = \frac{29}{11}$

Therefore, the solution to the system of equations is $\left(\frac{17}{11}, \frac{29}{11}\right)$.

Final Thoughts

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The substitution method is a useful technique for solving systems of linear equations. However, it is not always possible to find a solution to a system of equations. In this case, the system of equations has no solution, which means that there is no value of $x$ that can satisfy both equations.

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The substitution method is a technique used to solve systems of linear equations by substituting one equation into the other. This method is particularly useful when one of the equations is already solved for one of the variables. In this article, we will answer some frequently asked questions about solving systems of equations using the substitution method.

Q: What is the substitution method?

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A: The substitution method is a technique used to solve systems of linear equations by substituting one equation into the other. This method is particularly useful when one of the equations is already solved for one of the variables.

Q: When is the substitution method useful?

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A: The substitution method is particularly useful when one of the equations is already solved for one of the variables. This makes it easier to substitute the expression for the variable into the other equation.

Q: How do I know which equation to substitute into the other?

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A: You can choose either equation to substitute into the other. However, it is often easier to substitute the equation that is already solved for one of the variables.

Q: What happens if the substitution method results in a contradiction?

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A: If the substitution method results in a contradiction, it means that there is no solution to the system of equations. This is because the two sides of the equation are not equal, and therefore, there is no value of the variable that can satisfy the equation.

Q: Can I use the substitution method to solve systems of equations with more than two variables?

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A: Yes, you can use the substitution method to solve systems of equations with more than two variables. However, it may be more difficult to solve the system of equations, and you may need to use other methods, such as the elimination method or the graphing method.

Q: How do I know if the substitution method is the best method to use?

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A: You can use the substitution method if one of the equations is already solved for one of the variables. If not, you may need to use other methods, such as the elimination method or the graphing method.

Q: Can I use the substitution method to solve systems of equations with fractions or decimals?

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A: Yes, you can use the substitution method to solve systems of equations with fractions or decimals. However, you may need to simplify the equations before substituting one equation into the other.

Q: How do I check my solution to make sure it is correct?

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A: You can check your solution by substituting the values of the variables back into the original equations. If the solution satisfies both equations, then it is correct.

Q: What are some common mistakes to avoid when using the substitution method?

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A: Some common mistakes to avoid when using the substitution method include:

• Not simplifying the equations before substituting one equation into the other
• Not checking the solution to make sure it is correct
• Not using the correct method to solve the system of equations

Q: Can I use the substitution method to solve systems of equations with non-linear equations?

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A: No, you cannot use the substitution method to solve systems of equations with non-linear equations. The substitution method is only used to solve systems of linear equations.

Q: How do I know if the substitution method is the best method to use for a system of equations with non-linear equations?

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A: If you have a system of equations with non-linear equations, you may need to use other methods, such as the graphing method or the numerical method.

Q: Can I use the substitution method to solve systems of equations with complex numbers?

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A: Yes, you can use the substitution method to solve systems of equations with complex numbers. However, you may need to simplify the equations before substituting one equation into the other.

Q: How do I check my solution to make sure it is correct when using complex numbers?

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A: You can check your solution by substituting the values of the variables back into the original equations. If the solution satisfies both equations, then it is correct.

Q: What are some common mistakes to avoid when using the substitution method with complex numbers?

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A: Some common mistakes to avoid when using the substitution method with complex numbers include:

• Not simplifying the equations before substituting one equation into the other
• Not checking the solution to make sure it is correct
• Not using the correct method to solve the system of equations

Q: Can I use the substitution method to solve systems of equations with matrices?

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A: Yes, you can use the substitution method to solve systems of equations with matrices. However, you may need to simplify the equations before substituting one equation into the other.

Q: How do I check my solution to make sure it is correct when using matrices?

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A: You can check your solution by substituting the values of the variables back into the original equations. If the solution satisfies both equations, then it is correct.

Q: What are some common mistakes to avoid when using the substitution method with matrices?

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A: Some common mistakes to avoid when using the substitution method with matrices include:

• Not simplifying the equations before substituting one equation into the other
• Not checking the solution to make sure it is correct
• Not using the correct method to solve the system of equations

Q: Can I use the substitution method to solve systems of equations with systems of inequalities?

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A: No, you cannot use the substitution method to solve systems of equations with systems of inequalities. The substitution method is only used to solve systems of linear equations.

Q: How do I know if the substitution method is the best method to use for a system of equations with systems of inequalities?

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A: If you have a system of equations with systems of inequalities, you may need to use other methods, such as the graphing method or the numerical method.

Q: Can I use the substitution method to solve systems of equations with systems of equations with multiple variables?

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A: Yes, you can use the substitution method to solve systems of equations with systems of equations with multiple variables. However, you may need to simplify the equations before substituting one equation into the other.

Q: How do I check my solution to make sure it is correct when using systems of equations with multiple variables?

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A: You can check your solution by substituting the values of the variables back into the original equations. If the solution satisfies both equations, then it is correct.

Q: What are some common mistakes to avoid when using the substitution method with systems of equations with multiple variables?

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A: Some common mistakes to avoid when using the substitution method with systems of equations with multiple variables include:

• Not simplifying the equations before substituting one equation into the other
• Not checking the solution to make sure it is correct
• Not using the correct method to solve the system of equations

Q: Can I use the substitution method to solve systems of equations with systems of equations with complex numbers?

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A: Yes, you can use the substitution method to solve systems of equations with systems of equations with complex numbers. However, you may need to simplify the equations before substituting one equation into the other.

Q: How do I check my solution to make sure it is correct when using systems of equations with complex numbers?

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A: You can check your solution by substituting the values of the variables back into the original equations. If the solution satisfies both equations, then it is correct.

Q: What are some common mistakes to avoid when using the substitution method with systems of equations with complex numbers?

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A: Some common mistakes to avoid when using the substitution method with systems of equations with complex numbers include:

• Not simplifying the equations before substituting one equation into the other
• Not checking the solution to make sure it is correct
• Not using the correct method to solve the system of equations

Q: Can I use the substitution method to solve systems of equations with systems of equations with matrices?

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A: Yes, you can use the substitution method to solve systems of equations with systems of equations with matrices. However, you may need to simplify the equations before substituting one equation into the other.

Q: How do I check my solution to make sure it is correct when using systems of equations with matrices?

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A: You can check your solution by substituting the values of the variables back into the original equations. If the solution satisfies both equations, then it is correct.

Q: What are some common mistakes to avoid when using the substitution method with systems of equations with matrices?

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A: Some common mistakes to avoid when using the substitution method with systems of equations with matrices include:

• Not simplifying the equations before substituting one equation into the other
• Not checking the solution to make sure it is correct
• Not using the correct method to solve the system of equations

Q: Can I use the substitution method to solve systems of equations with systems of equations with systems of inequalities?

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A: No, you cannot use the substitution method to solve systems of equations with systems of equations with systems of inequalities. The substitution method is only used to solve systems of linear equations.

Q: How do I know if the substitution method is the best method to use for a system of equations with systems of equations with systems of inequalities?

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A: If you have a system of equations with systems of equations with systems of inequalities, you may need to use other methods, such as the graphing method or the numerical method.

Q: Can I use the substitution method to solve systems of equations with systems of equations with multiple variables and complex numbers?

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A: Yes, you can use the substitution method to solve systems of equations with systems of equations with multiple variables and complex numbers. However