The first equation can be graphed

Oct 13, 2025 by magicalceylontours.com 66 views

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Introduction

Solving a system of equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. 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.

What is the Substitution Method?

The substitution method is a technique used to solve a system of equations by substituting one equation into the other. This method is useful when one of the equations can be easily solved for one variable in terms of the other. The substitution method involves the following steps:

Solve one of the equations for one variable in terms of the other.
Substitute the expression obtained in step 1 into the other equation.
Solve the resulting equation for the remaining variable.
Back-substitute the value obtained in step 3 into one of the original equations to find the value of the other variable.

Example 1: Solving a System of Equations Using Substitution

Let's consider the following system of equations:

{ \begin{cases} 2x + 6y = 10, \\ 4x - y = 7 \end{cases} \}

To solve this system using the substitution method, we can start by solving the second equation for $y$ in terms of $x$ .

Step 1: Solve the Second Equation for y

We can solve the second equation for $y$ by isolating $y$ on one side of the equation.

$4x - y = 7$

$-y = 7 - 4x$

$y = 4x - 7$

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

Now that we have an expression for $y$ in terms of $x$ , we can substitute this expression into the first equation.

$2x + 6y = 10$

$2x + 6(4x - 7) = 10$

Step 3: Solve the Resulting Equation for x

We can now solve the resulting equation for $x$ .

$2x + 24x - 42 = 10$

$26x - 42 = 10$

$26x = 52$

$x = 2$

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

Now that we have the value of $x$ , we can back-substitute this value into one of the original equations to find the value of $y$ .

$y = 4x - 7$

$y = 4(2) - 7$

$y = 8 - 7$

$y = 1$

Therefore, the solution to the system of equations is $x = 2$ and $y = 1$ .

Example 2: Solving a System of Equations Using Substitution

Let's consider the following system of equations:

{ \begin{cases} 7x - 4y = 2, \\ 5x + 11y = 43 \end{cases} \}

To solve this system using the substitution method, we can start by solving the first equation for $y$ in terms of $x$ .

Step 1: Solve the First Equation for y

We can solve the first equation for $y$ by isolating $y$ on one side of the equation.

$7x - 4y = 2$

$-4y = 2 - 7x$

$y = \frac{7x - 2}{4}$

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

Now that we have an expression for $y$ in terms of $x$ , we can substitute this expression into the second equation.

$5x + 11y = 43$

$5x + 11\left(\frac{7x - 2}{4}\right) = 43$

Step 3: Solve the Resulting Equation for x

We can now solve the resulting equation for $x$ .

$5x + \frac{77x - 22}{4} = 43$

$20x + 77x - 22 = 172$

$97x - 22 = 172$

$97x = 194$

$x = 2$

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

Now that we have the value of $x$ , we can back-substitute this value into one of the original equations to find the value of $y$ .

$y = \frac{7x - 2}{4}$

$y = \frac{7(2) - 2}{4}$

$y = \frac{14 - 2}{4}$

$y = \frac{12}{4}$

$y = 3$

Therefore, the solution to the system of equations is $x = 2$ and $y = 3$ .

Conclusion

In this article, we have discussed the substitution method for solving a system of equations. We have provided two examples of solving a system of equations using the substitution method. The substitution method is a useful technique for solving a system of equations when one of the equations can be easily solved for one variable in terms of the other. By following the steps outlined in this article, you can use the substitution method to solve a system of equations and find the values of the variables.

References

[1] "Algebra and Trigonometry" by James Stewart
[2] "College Algebra" by James Stewart
[3] "Linear Algebra and Its Applications" by Gilbert Strang

Future Work

In the future, we plan to explore other methods for solving a system of equations, including the elimination method and the graphing method. We also plan to provide more examples of solving a system of equations using the substitution method.

Acknowledgments

We would like to thank our colleagues and mentors for their support and guidance in writing this article. We would also like to thank the readers for their interest in this topic.

Glossary

System of Equations: A set of two or more equations that are solved simultaneously to find the values of the variables.
Substitution Method: A technique used to solve a system of equations by substituting one equation into the other.
Elimination Method: A technique used to solve a system of equations by eliminating one variable by adding or subtracting the equations.
Graphing Method: A technique used to solve a system of equations by graphing the equations on a coordinate plane and finding the point of intersection.

Appendices

A.1: Derivation of the Substitution Method

The substitution method can be derived by solving one of the equations for one variable in terms of the other and then substituting this expression into the other equation.

A.2: Example of Solving a System of Equations Using the Elimination Method

Let's consider the following system of equations:

{ \begin{cases} 2x + 3y = 7, \\ x - 2y = -3 \end{cases} \}

To solve this system using the elimination method, we can start by multiplying the second equation by 2 to make the coefficients of $x$ in both equations equal.

$2(x - 2y) = 2(-3)$

$2x - 4y = -6$

Now that we have two equations with the same coefficient for $x$ , we can add the equations to eliminate the variable $x$ .

$(2x + 3y) + (2x - 4y) = 7 + (-6)$

$4x - y = 1$

We can now solve the resulting equation for $y$ .

$-y = 1 - 4x$

$y = 4x - 1$

Now that we have an expression for $y$ in terms of $x$ , we can substitute this expression into one of the original equations to find the value of $x$ .

$2x + 3y = 7$

$2x + 3(4x - 1) = 7$

$2x + 12x - 3 = 7$

$14x - 3 = 7$

$14x = 10$

$x = \frac{5}{7}$

Now that we have the value of $x$ , we can back-substitute this value into one of the original equations to find the value of $y$ .

$y = 4x - 1$

$y = 4\left(\frac{5}{7}\right) - 1$

$y = \frac{20}{7} - 1$

$y = \frac{20 - 7}{7}$

$y = \frac{13}{7}$

Therefore, the solution to the system of equations is $x = \frac{5}{7}$ and $y = \frac{13}{7}$ .

A.3: Example of Solving a System of Equations Using the Graphing Method

Let's consider the following system of equations:

{ \begin{cases} x + 2y = 4, \\ 2x - 3y = -1 \end{cases} \}

To solve this system using the graphing method, we can start by graphing the equations on a coordinate plane.

The first equation can be graphed

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Introduction

Solving systems of equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In our previous article, we discussed the substitution method for solving systems of equations. In this article, we will provide a Q&A section to help you better understand the substitution method and how to apply it to solve systems of equations.

Q1: What is the substitution method?

A1: The substitution method is a technique used to solve a system of equations by substituting one equation into the other. This method is useful when one of the equations can be easily solved for one variable in terms of the other.

Q2: How do I choose which equation to solve for first?

A2: When choosing which equation to solve for first, look for an equation that is easy to solve for one variable in terms of the other. For example, if one equation has a coefficient of 1 for one variable, it may be easier to solve for that variable first.

Q3: What if I get stuck during the substitution process?

A3: If you get stuck during the substitution process, try to simplify the equation by combining like terms or factoring out common factors. If you are still having trouble, try to visualize the problem and see if you can find a different approach.

Q4: Can I use the substitution method with systems of equations that have more than two variables?

A4: Yes, you can use the substitution method with systems of equations that have more than two variables. However, the process may be more complicated and may require more steps.

Q5: How do I know if the substitution method is the best approach for a particular system of equations?

A5: To determine if the substitution method is the best approach for a particular system of equations, try to solve one of the equations for one variable in terms of the other. If this is possible, then the substitution method may be a good approach.

Q6: What if I get a system of equations with no solution?

A6: If you get a system of equations with no solution, it means that the equations are inconsistent and cannot be solved simultaneously. This can happen if the equations are contradictory or if one equation is a multiple of the other.

Q7: Can I use the substitution method with systems of equations that have fractions or decimals?

A7: Yes, you can use the substitution method with systems of equations that have fractions or decimals. However, you may need to simplify the equations by combining like terms or factoring out common factors.

Q8: How do I check my solution to make sure it is correct?

A8: To check your solution, substitute the values of the variables back into the original equations and make sure that they are true. If the equations are true, then your solution is correct.

Q9: Can I use the substitution method with systems of equations that have absolute values or square roots?

A9: 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 equations by combining like terms or factoring out common factors.

Q10: Where can I find more practice problems to help me improve my skills?

A10: You can find more practice problems in your textbook or online resources such as Khan Academy, Mathway, or Wolfram Alpha.

Conclusion

In this Q&A article, we have provided answers to common questions about the substitution method for solving systems of equations. We hope that this article has helped you better understand the substitution method and how to apply it to solve systems of equations.

References

[1] "Algebra and Trigonometry" by James Stewart
[2] "College Algebra" by James Stewart
[3] "Linear Algebra and Its Applications" by Gilbert Strang

Future Work

In the future, we plan to provide more Q&A articles on various topics in mathematics, including algebra, geometry, and calculus.

Acknowledgments

We would like to thank our colleagues and mentors for their support and guidance in writing this article. We would also like to thank the readers for their interest in this topic.

Glossary

System of Equations: A set of two or more equations that are solved simultaneously to find the values of the variables.
Substitution Method: A technique used to solve a system of equations by substituting one equation into the other.
Elimination Method: A technique used to solve a system of equations by eliminating one variable by adding or subtracting the equations.
Graphing Method: A technique used to solve a system of equations by graphing the equations on a coordinate plane and finding the point of intersection.

Appendices

A.1: Derivation of the Substitution Method

The substitution method can be derived by solving one of the equations for one variable in terms of the other and then substituting this expression into the other equation.

A.2: Example of Solving a System of Equations Using the Elimination Method

Let's consider the following system of equations:

{ \begin{cases} 2x + 3y = 7, \\ x - 2y = -3 \end{cases} \}

To solve this system using the elimination method, we can start by multiplying the second equation by 2 to make the coefficients of $x$ in both equations equal.

$2(x - 2y) = 2(-3)$

$2x - 4y = -6$

Now that we have two equations with the same coefficient for $x$ , we can add the equations to eliminate the variable $x$ .

$(2x + 3y) + (2x - 4y) = 7 + (-6)$

$4x - y = 1$

We can now solve the resulting equation for $y$ .

$-y = 1 - 4x$

$y = 4x - 1$

Now that we have an expression for $y$ in terms of $x$ , we can substitute this expression into one of the original equations to find the value of $x$ .

$2x + 3y = 7$

$2x + 3(4x - 1) = 7$

$2x + 12x - 3 = 7$

$14x - 3 = 7$

$14x = 10$

$x = \frac{5}{7}$

Now that we have the value of $x$ , we can back-substitute this value into one of the original equations to find the value of $y$ .

$y = 4x - 1$

$y = 4\left(\frac{5}{7}\right) - 1$

$y = \frac{20}{7} - 1$

$y = \frac{20 - 7}{7}$

$y = \frac{13}{7}$

Therefore, the solution to the system of equations is $x = \frac{5}{7}$ and $y = \frac{13}{7}$ .

A.3: Example of Solving a System of Equations Using the Graphing Method

Let's consider the following system of equations:

{ \begin{cases} x + 2y = 4, \\ 2x - 3y = -1 \end{cases} \}

To solve this system using the graphing method, we can start by graphing the equations on a coordinate plane.

The first equation can be graphed