Which Equation Is Part Of Solving The System By Substitution?$\[ \begin{cases} x + Y = 11 \\ 4x^2 - 3y^2 = 8 \end{cases} \\]A. \[$4(y + 11)^2 - 3y^2 = 8\$\]B. \[$4(11 - Y)^2 - 3y^2 = 8\$\]C. \[$4(y - 11)^2 - 3y^2 =

Introduction

Solving systems of equations is a fundamental concept in mathematics, and one of the most effective methods is substitution. This technique involves solving one equation for a variable and then substituting that expression into the other equation. In this article, we will explore the process of solving a system of equations by substitution and provide a step-by-step guide on how to choose the correct equation to substitute.

What is Substitution?

Substitution is a method of solving systems of equations by solving one equation for a variable and then substituting that expression into the other equation. This technique is useful when one of the equations is linear and the other is quadratic or when one of the equations is easier to solve than the other.

Choosing the Correct Equation to Substitute

When solving a system of equations by substitution, we need to choose the correct equation to substitute. The correct equation to substitute is the one that is easier to solve or the one that has a variable that is already isolated. In the given system of equations:

{ \begin{cases} x + y = 11 \\ 4x^2 - 3y^2 = 8 \end{cases} \}

We need to choose the correct equation to substitute. Let's analyze the options:

Option A: $4(y + 11)^2 - 3y^2 = 8$

This equation is obtained by substituting $x = 11 - y$ into the second equation. However, this substitution is not correct because it does not isolate the variable $y$.

Option B: $4(11 - y)^2 - 3y^2 = 8$

This equation is obtained by substituting $x = 11 - y$ into the second equation. This substitution is correct because it isolates the variable $y$.

Option C: $4(y - 11)^2 - 3y^2 = 8$

This equation is obtained by substituting $x = y - 11$ into the second equation. However, this substitution is not correct because it does not isolate the variable $y$.

Step-by-Step Guide to Solving the System by Substitution

Now that we have chosen the correct equation to substitute, let's follow the step-by-step guide to solve the system by substitution:

Step 1: Solve the First Equation for x

The first equation is $x + y = 11$. We can solve this equation for $x$ by subtracting $y$ from both sides:

$x = 11 - y$

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

We can substitute the expression for $x$ into the second equation:

$4(11 - y)^2 - 3y^2 = 8$

Step 3: Expand and Simplify the Equation

We can expand and simplify the equation:

$4(121 - 22y + y^2) - 3y^2 = 8$

$484 - 88y + 4y^2 - 3y^2 = 8$

$484 - 88y + y^2 = 8$

Step 4: Rearrange the Equation

We can rearrange the equation to get a quadratic equation in terms of $y$:

$y^2 - 88y + 476 = 0$

Step 5: Solve the Quadratic Equation

We can solve the quadratic equation using the quadratic formula:

$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$y = \frac{88 \pm \sqrt{7744 - 1904}}{2}$

$y = \frac{88 \pm \sqrt{5840}}{2}$

$y = \frac{88 \pm 76.4}{2}$

$y = 82.2$ or $y = 5.8$

Step 6: Find the Value of x

We can find the value of $x$ by substituting the value of $y$ into the first equation:

$x + y = 11$

$x + 82.2 = 11$

$x = -71.2$

$x + 5.8 = 11$

$x = 5.2$

Step 7: Check the Solutions

We can check the solutions by substituting the values of $x$ and $y$ into both equations:

$x + y = 11$

$-71.2 + 82.2 = 11$

$11.0 = 11$

$x + y = 11$

$5.2 + 5.8 = 11$

$11.0 = 11$

The solutions are $x = -71.2$ and $y = 82.2$ and $x = 5.2$ and $y = 5.8$.

Conclusion

Q: What is substitution in solving systems of equations?

A: Substitution is a method of solving systems of equations by solving one equation for a variable and then substituting that expression into the other equation.

Q: Why is substitution useful?

A: Substitution is useful when one of the equations is linear and the other is quadratic or when one of the equations is easier to solve than the other.

Q: How do I choose the correct equation to substitute?

A: To choose the correct equation to substitute, you need to analyze the equations and determine which one is easier to solve or which one has a variable that is already isolated.

Q: What are the steps to solve a system of equations by substitution?

A: The steps to solve a system of equations by substitution are:

1. Solve the first equation for x.
2. Substitute the expression for x into the second equation.
3. Expand and simplify the equation.
4. Rearrange the equation to get a quadratic equation in terms of y.
5. Solve the quadratic equation using the quadratic formula.
6. Find the value of x by substituting the value of y into the first equation.
7. Check the solutions by substituting the values of x and y into both equations.

Q: What are some common mistakes to avoid when solving systems of equations by substitution?

A: Some common mistakes to avoid when solving systems of equations by substitution are:

• Not choosing the correct equation to substitute.
• Not expanding and simplifying the equation correctly.
• Not rearranging the equation to get a quadratic equation in terms of y.
• Not solving the quadratic equation correctly.
• Not checking the solutions by substituting the values of x and y into both equations.

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

A: Yes, you can use substitution to solve systems of equations with more than two variables. However, you need to be careful when choosing the correct equation to substitute and when expanding and simplifying the equation.

Q: What are some real-world applications of solving systems of equations by substitution?

A: Some real-world applications of solving systems of equations by substitution include:

• Finding the intersection points of two lines or curves.
• Determining the optimal solution to a problem that involves multiple variables.
• Solving problems in physics, engineering, and economics that involve systems of equations.

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

A: Yes, you can use substitution to solve systems of equations with non-linear equations. However, you need to be careful when choosing the correct equation to substitute and when expanding and simplifying the equation.

Q: What are some tips for solving systems of equations by substitution?

A: Some tips for solving systems of equations by substitution include:

• Choose the correct equation to substitute carefully.
• Expand and simplify the equation correctly.
• Rearrange the equation to get a quadratic equation in terms of y.
• Solve the quadratic equation correctly.
• Check the solutions by substituting the values of x and y into both equations.

Conclusion

Solving systems of equations by substitution is a powerful technique that can be used to solve systems of linear and quadratic equations. By following the steps and tips outlined in this article, you can solve systems of equations by substitution and find the values of the variables.