Use The Substitution Method To Solve The Following System Of Equations:${ \begin{array}{l} 9y^2 + 8x = 12 \ 2x + 3y = 4 \end{array} }$

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 the substitution method, the elimination method, and the graphing method. In this article, we will focus on the substitution method, which involves solving one equation for one variable and then substituting that expression into the other equation.

What is the Substitution Method?

The substitution method is a technique used to solve a system of equations by substituting the expression for one variable from one equation into the other equation. This method is useful when one equation is linear and the other equation is quadratic or higher degree. The substitution method involves the following steps:

1. Solve one equation for one variable.
2. Substitute the expression for that variable into the other equation.
3. Solve the resulting equation for the other variable.
4. Back-substitute the value of the second variable into one of the original equations to find the value of the first variable.

Step-by-Step Solution

Let's use the substitution method to solve the following system of equations:

{ \begin{array}{l} 9y^2 + 8x = 12 \\ 2x + 3y = 4 \end{array} \}

Step 1: Solve the Second Equation for x

We will solve the second equation for x:

${ 2x + 3y = 4 }$

Subtracting 3y from both sides gives:

${ 2x = 4 - 3y }$

Dividing both sides by 2 gives:

${ x = \frac{4 - 3y}{2} }$

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

We will substitute the expression for x into the first equation:

${ 9y^2 + 8x = 12 }$

Substituting x = (4 - 3y)/2 gives:

${ 9y^2 + 8\left(\frac{4 - 3y}{2}\right) = 12 }$

Simplifying the equation gives:

${ 9y^2 + 4(4 - 3y) = 12 }$

Expanding the equation gives:

${ 9y^2 + 16 - 12y = 12 }$

Subtracting 16 from both sides gives:

${ 9y^2 - 12y - 4 = 0 }$

Step 3: Solve the Quadratic Equation

We will solve the quadratic equation using the quadratic formula:

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

In this case, a = 9, b = -12, and c = -4. Plugging these values into the formula gives:

${ y = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(9)(-4)}}{2(9)} }$

Simplifying the equation gives:

${ y = \frac{12 \pm \sqrt{144 + 144}}{18} }$

Simplifying further gives:

${ y = \frac{12 \pm \sqrt{288}}{18} }$

Simplifying the square root gives:

${ y = \frac{12 \pm 12\sqrt{2}}{18} }$

Simplifying further gives:

${ y = \frac{2 \pm 2\sqrt{2}}{3} }$

Step 4: Back-Substitute the Value of y into One of the Original Equations

We will back-substitute the value of y into the second original equation:

${ 2x + 3y = 4 }$

Substituting y = (2 + 2√2)/3 gives:

${ 2x + 3\left(\frac{2 + 2\sqrt{2}}{3}\right) = 4 }$

Simplifying the equation gives:

${ 2x + 2 + 2\sqrt{2} = 4 }$

Subtracting 2 + 2√2 from both sides gives:

${ 2x = 2 - 2\sqrt{2} }$

Dividing both sides by 2 gives:

${ x = \frac{2 - 2\sqrt{2}}{2} }$

Simplifying further gives:

${ x = 1 - \sqrt{2} }$

Conclusion

In this article, we used the substitution method to solve a system of equations. We solved one equation for one variable and then substituted that expression into the other equation. We then solved the resulting equation for the other variable and back-substituted the value of the second variable into one of the original equations to find the value of the first variable. The substitution method is a useful technique for solving systems of equations, especially when one equation is linear and the other equation is quadratic or higher degree.

Example Use Cases

The substitution method can be used to solve a wide range of systems of equations, including:

• Linear systems of equations
• Quadratic systems of equations
• Higher-degree systems of equations
• Systems of equations with multiple variables

Tips and Tricks

Here are some tips and tricks for using the substitution method:

• Make sure to solve one equation for one variable before substituting that expression into the other equation.
• Use the quadratic formula to solve quadratic equations.
• Back-substitute the value of the second variable into one of the original equations to find the value of the first variable.
• Check your solutions by plugging them back into the original equations.

Conclusion

Q: What is the substitution method?

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

Q: When should I use the substitution method?

A: You should use the substitution method when one equation is linear and the other equation is quadratic or higher degree.

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

A: You should solve the equation that is easiest to solve for first. If one equation is linear and the other equation is quadratic, it is usually easier to solve the linear equation for first.

Q: What if I get a quadratic equation that I don't know how to solve?

A: If you get a quadratic equation that you don't know how to solve, you can use the quadratic formula to solve it. The quadratic formula is:

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

Q: What if I get a negative value for y?

A: If you get a negative value for y, it means that the solution is not valid. You should go back and check your work to make sure that you made no mistakes.

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

A: Yes, you can use the substitution method to solve a system of equations with multiple variables. However, it may be more difficult to solve the system of equations with multiple variables.

Q: What if I get a system of equations with no solution?

A: If you get a system of equations with no solution, it means that the system of equations is inconsistent. This means that the equations are contradictory, and there is no solution.

Q: Can I use the substitution method to solve a system of equations with infinitely many solutions?

A: Yes, you can use the substitution method to solve a system of equations with infinitely many solutions. However, it may be more difficult to solve the system of equations with infinitely many solutions.

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 using the quadratic formula to solve quadratic equations.
• Not back-substituting the value of the second variable into one of the original equations to find the value of the first variable.
• Not checking your solutions by plugging them back into the original equations.

Q: How can I practice using the substitution method?

A: You can practice using the substitution method by working through examples and exercises. You can also try using the substitution method to solve real-world problems.

Q: What are some real-world applications of the substitution method?

A: Some real-world applications of the substitution method include:

• Solving systems of equations in physics and engineering.
• Solving systems of equations in economics and finance.
• Solving systems of equations in computer science and programming.

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 solve for the values of the variables. The substitution method can be used to solve a wide range of systems of equations, including linear, quadratic, and higher-degree systems. With practice and patience, you can become proficient in using the substitution method to solve systems of equations.