1. Solve The System Of Equations By Substitution.$\[ \begin{array}{l} y = 6x - 11 \\ -2x - 3y = -7 \end{array} \\]A. \[$(2, 1)\$\] B. \[$(-2, -23)\$\] C. \[$(-23, -2)\$\] D. \[$(1, 2)\$\]

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Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. There are several methods to solve a system of equations, including substitution, elimination, and graphing. In this article, we will focus on solving a system of equations using the substitution method.

The Substitution Method

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.

Step 1: Solve One Equation for One Variable

We are given two equations:

y=6x−11−2x−3y=−7\begin{array}{l} y = 6x - 11 \\ -2x - 3y = -7 \end{array}

We can solve the first equation for y:

y=6x−11y = 6x - 11

Step 2: Substitute the Expression into the Other Equation

Now, we can substitute the expression for y into the second equation:

−2x−3(6x−11)=−7-2x - 3(6x - 11) = -7

Step 3: Simplify the Equation

Next, we can simplify the equation by distributing the -3 to the terms inside the parentheses:

−2x−18x+33=−7-2x - 18x + 33 = -7

Step 4: Combine Like Terms

Now, we can combine like terms:

−20x+33=−7-20x + 33 = -7

Step 5: Isolate the Variable

Next, we can isolate the variable x by subtracting 33 from both sides of the equation:

−20x=−40-20x = -40

Step 6: Solve for the Variable

Finally, we can solve for x by dividing both sides of the equation by -20:

x=2x = 2

Step 7: 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. We will use the first equation:

y=6x−11y = 6x - 11

y=6(2)−11y = 6(2) - 11

y=12−11y = 12 - 11

y=1y = 1

Conclusion

Therefore, the solution to the system of equations is (2, 1).

Answer

The correct answer is:

A. (2,1)(2, 1)

Explanation

The substitution method is a useful technique for solving systems of equations. By solving one equation for one variable and then substituting that expression into the other equation, we can find the values of the variables. In this case, we solved the first equation for y and then substituted that expression into the second equation. We then simplified the equation, combined like terms, isolated the variable, and solved for the variable. Finally, we found the value of the other variable by substituting the value of x into one of the original equations.

Example

Here is an example of how to use the substitution method to solve a system of equations:

x+y=42x−3y=−3\begin{array}{l} x + y = 4 \\ 2x - 3y = -3 \end{array}

We can solve the first equation for x:

x=4−yx = 4 - y

We can then substitute this expression into the second equation:

2(4−y)−3y=−32(4 - y) - 3y = -3

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

8−2y−3y=−38 - 2y - 3y = -3

We can combine like terms:

8−5y=−38 - 5y = -3

We can isolate the variable y by subtracting 8 from both sides of the equation:

−5y=−11-5y = -11

We can solve for y by dividing both sides of the equation by -5:

y=115y = \frac{11}{5}

We can then find the value of x by substituting y into one of the original equations. We will use the first equation:

x=4−yx = 4 - y

x=4−115x = 4 - \frac{11}{5}

x=95x = \frac{9}{5}

Therefore, the solution to the system of equations is (95,115)\left(\frac{9}{5}, \frac{11}{5}\right).

Tips and Tricks

Here are some tips and tricks for using the substitution method to solve systems of equations:

  • Make sure to solve one equation for one variable before substituting that expression into the other equation.
  • Simplify the equation by combining like terms and isolating the variable.
  • Check your work by plugging the values of the variables back into the original equations.
  • Use the substitution method when one of the equations is already solved for one variable.

Conclusion

In conclusion, the substitution method is a useful technique for solving systems of equations. By solving one equation for one variable and then substituting that expression into the other equation, we can find the values of the variables. With practice and patience, you can become proficient in using the substitution method to solve systems of equations.

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Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of equations by solving one equation for one variable and then substituting that expression into the other equation.

Q: When should I use the substitution method?

A: You should use the substitution method when one of the equations is already solved for one variable. This method is useful when you have a simple equation that can be easily solved for one variable.

Q: How do I know which equation to solve for first?

A: You should solve the equation that is easiest to solve for one variable. If one equation is already solved for one variable, you can use that equation as the first equation to solve for.

Q: What if I have two equations with two variables and neither equation is easily solvable for one variable?

A: In this case, you can use the elimination method or the graphing method to solve the system of equations.

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

A: No, the substitution method is typically used for systems of equations with two variables. If you have a system of equations with more than two variables, you may need to use a different method, such as the elimination method or the graphing method.

Q: How do I check my work when using the substitution method?

A: To check your work, you should plug the values of the variables back into the original equations and make sure that they are true. If the values of the variables do not satisfy both equations, then you have made an error and need to re-solve the system of equations.

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

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

  • Not solving one equation for one variable before substituting that expression into the other equation.
  • Not simplifying the equation by combining like terms and isolating the variable.
  • Not checking your work by plugging the values of the variables back into the original equations.

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

A: Yes, you can use the substitution method with systems of equations that have fractions or decimals. However, you may need to simplify the equation by combining like terms and isolating the variable.

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

A: You should use the substitution method when one of the equations is already solved for one variable and the other equation is simple. If the equations are complex or neither equation is easily solvable for one variable, you may need to use a different method, such as the elimination method or the graphing method.

Q: Can I use the substitution method with systems of equations that have absolute values or square roots?

A: Yes, you can use the substitution method with systems of equations that have absolute values or square roots. However, you may need to simplify the equation by combining like terms and isolating the variable.

Conclusion

In conclusion, the substitution method is a useful technique for solving systems of equations. By solving one equation for one variable and then substituting that expression into the other equation, you can find the values of the variables. With practice and patience, you can become proficient in using the substitution method to solve systems of equations.

Tips and Tricks

Here are some tips and tricks for using the substitution method to solve systems of equations:

  • Make sure to solve one equation for one variable before substituting that expression into the other equation.
  • Simplify the equation by combining like terms and isolating the variable.
  • Check your work by plugging the values of the variables back into the original equations.
  • Use the substitution method when one of the equations is already solved for one variable and the other equation is simple.

Practice Problems

Here are some practice problems to help you become proficient in using the substitution method to solve systems of equations:

  1. x+y=42x−3y=−3\begin{array}{l} x + y = 4 \\ 2x - 3y = -3 \end{array}

    Solve the system of equations using the substitution method.

  2. x−y=23x+2y=7\begin{array}{l} x - y = 2 \\ 3x + 2y = 7 \end{array}

    Solve the system of equations using the substitution method.

  3. 2x+3y=5x−2y=−3\begin{array}{l} 2x + 3y = 5 \\ x - 2y = -3 \end{array}

    Solve the system of equations using the substitution method.

Conclusion

In conclusion, the substitution method is a useful technique for solving systems of equations. By solving one equation for one variable and then substituting that expression into the other equation, you can find the values of the variables. With practice and patience, you can become proficient in using the substitution method to solve systems of equations.