Use The Method Of Elimination To Solve The Following System Of Equations. If The System Is Dependent, Express The Solution Set In Terms Of One Of The Variables. Leave All Fractional Answers In Fraction Form.$[ \begin{cases} -2x + 5y = -21 \ 8x -

Introduction

The method of elimination is a powerful technique used to solve systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, allowing us to solve for the other variable. In this article, we will use the method of elimination to solve a system of two linear equations. We will also discuss how to express the solution set in terms of one of the variables if the system is dependent.

The Method of Elimination

The method of elimination involves adding or subtracting equations to eliminate one of the variables. To do this, we need to have the coefficients of one of the variables in both equations to be the same, but with opposite signs. Once we have done this, we can add or subtract the equations to eliminate the variable.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to have the coefficients of that variable in both equations to be the same, but with opposite signs. To do this, we can multiply one or both of the equations by necessary multiples.

Let's consider the following system of equations:

$${ \begin{cases} -2x + 5y = -21 \\ 8x - 3y = 12 \end{cases} }$$

We can see that the coefficients of $x$ in both equations are not the same. To eliminate $x$, we can multiply the first equation by $-4$ and the second equation by $1$. This will give us:

$${ \begin{cases} 8x - 20y = 84 \\ 8x - 3y = 12 \end{cases} }$$

Step 2: Add or Subtract the Equations

Now that we have the coefficients of $x$ in both equations to be the same, we can add or subtract the equations to eliminate $x$. Let's add the two equations:

$${ (8x - 20y) + (8x - 3y) = 84 + 12 }$$

Simplifying the equation, we get:

$${ 16x - 23y = 96 }$$

Step 3: Solve for One Variable

Now that we have eliminated $x$, we can solve for $y$. We can do this by isolating $y$ in the equation:

$${ 16x - 23y = 96 }$$

We can add $23y$ to both sides of the equation to get:

$${ 16x = 96 + 23y }$$

Now, we can divide both sides of the equation by $16$ to get:

$${ x = \frac{96 + 23y}{16} }$$

Step 4: Substitute the Expression for One Variable into the Other Equation

Now that we have an expression for $x$ in terms of $y$, we can substitute this expression into the other equation to solve for $y$. Let's substitute the expression for $x$ into the first equation:

$${ -2\left(\frac{96 + 23y}{16}\right) + 5y = -21 }$$

Simplifying the equation, we get:

$${ -\frac{192 + 46y}{16} + 5y = -21 }$$

Multiplying both sides of the equation by $16$, we get:

$${ -192 - 46y + 80y = -336 }$$

Simplifying the equation, we get:

$${ 34y = 144 }$$

Dividing both sides of the equation by $34$, we get:

$${ y = \frac{144}{34} }$$

Simplifying the fraction, we get:

$${ y = \frac{72}{17} }$$

Step 5: Substitute the Value of One Variable into the Expression for the Other Variable

Now that we have a value for $y$, we can substitute this value into the expression for $x$ to find the value of $x$. Let's substitute the value of $y$ into the expression for $x$:

$${ x = \frac{96 + 23\left(\frac{72}{17}\right)}{16} }$$

Simplifying the expression, we get:

$${ x = \frac{96 + \frac{1656}{17}}{16} }$$

Multiplying the numerator and denominator by $17$, we get:

$${ x = \frac{1632 + 1656}{272} }$$

Simplifying the expression, we get:

$${ x = \frac{3292}{272} }$$

Simplifying the fraction, we get:

$${ x = \frac{3292}{272} }$$

Conclusion

In this article, we used the method of elimination to solve a system of two linear equations. We multiplied the equations by necessary multiples, added or subtracted the equations to eliminate one of the variables, and solved for the other variable. We also expressed the solution set in terms of one of the variables if the system was dependent. The final answer is:

$${ \begin{cases} x = \frac{3292}{272} \\ y = \frac{72}{17} \end{cases} }$$

Discussion

The method of elimination is a powerful technique used to solve systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, allowing us to solve for the other variable. In this article, we used the method of elimination to solve a system of two linear equations. We multiplied the equations by necessary multiples, added or subtracted the equations to eliminate one of the variables, and solved for the other variable. We also expressed the solution set in terms of one of the variables if the system was dependent.

References

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Introduction to Linear Algebra" by Jim Hefferon

Keywords

• Method of elimination
• System of linear equations
• Linear algebra
• Mathematics
• Elimination method
• Linear equations
• Algebra
• Mathematics
  Frequently Asked Questions (FAQs) About Solving Systems of Equations Using the Method of Elimination =============================================================================================

Q: What is the method of elimination?

A: The method of elimination is a technique used to solve systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, allowing us to solve for the other variable.

Q: How do I know which variable to eliminate first?

A: To determine which variable to eliminate first, you need to look at the coefficients of the variables in both equations. If the coefficients of one variable are the same, but with opposite signs, you can eliminate that variable. If not, you can multiply one or both of the equations by necessary multiples to make the coefficients the same.

Q: What if the system of equations has more than two variables?

A: If the system of equations has more than two variables, you can use the method of elimination to eliminate one variable at a time. You can start by eliminating one variable, and then use the resulting equation to eliminate another variable.

Q: Can I use the method of elimination to solve a system of equations with fractions?

A: Yes, you can use the method of elimination to solve a system of equations with fractions. However, you need to be careful when multiplying or dividing fractions to avoid errors.

Q: What if the system of equations has no solution?

A: If the system of equations has no solution, it means that the equations are inconsistent. In this case, the method of elimination will not work, and you will need to use a different method to solve the system.

Q: What if the system of equations has infinitely many solutions?

A: If the system of equations has infinitely many solutions, it means that the equations are dependent. In this case, the method of elimination will not work, and you will need to use a different method to solve the system.

Q: Can I use the method of elimination to solve a system of equations with decimals?

A: Yes, you can use the method of elimination to solve a system of equations with decimals. However, you need to be careful when multiplying or dividing decimals to avoid errors.

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

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

• Not checking if the equations are consistent before using the method of elimination
• Not multiplying or dividing fractions correctly
• Not checking if the resulting equation is true before solving for the variable
• Not using the correct method to solve the system of equations

Q: How do I know if the system of equations is consistent or inconsistent?

A: To determine if the system of equations is consistent or inconsistent, you need to check if the equations have a solution. If the equations have a solution, the system is consistent. If the equations do not have a solution, the system is inconsistent.

Q: Can I use the method of elimination to solve a system of equations with absolute values?

A: Yes, you can use the method of elimination to solve a system of equations with absolute values. However, you need to be careful when dealing with absolute values to avoid errors.

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

A: Some real-world applications of the method of elimination 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
• Solving systems of equations in mathematics and statistics

Conclusion

In this article, we have answered some frequently asked questions about solving systems of equations using the method of elimination. We have discussed how to determine which variable to eliminate first, how to handle systems of equations with fractions, decimals, and absolute values, and how to avoid common mistakes when using the method of elimination. We have also discussed some real-world applications of the method of elimination.