Solve The System By Substitution.$\[ \begin{cases} 7x + 5y = -39 \\ x = 3y - 13 \end{cases} \\]Options:- One Or More Solutions: $\square$- No Solution- Infinite Number Of Solutions

Feb 26, 2025 by ADMIN 185 views

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Introduction

In this article, we will explore the method of substitution to solve a system of linear equations. The system consists of two equations with two variables, and we will use the substitution method to find the solution. This method involves solving one equation for one variable and then substituting that expression into the other equation.

The System of Equations

The system of equations is given as:

{ \begin{cases} 7x + 5y = -39 \\ x = 3y - 13 \end{cases} \}

Step 1: Solve the Second Equation for x

We will start by solving the second equation for x. This will give us an expression for x in terms of y.

{ x = 3y - 13 \}

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

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

{ 7(3y - 13) + 5y = -39 \}

Step 3: Simplify the Equation

We will now simplify the equation by distributing the 7 and combining like terms.

{ 21y - 91 + 5y = -39 \}

{ 26y - 91 = -39 \}

Step 4: Add 91 to Both Sides of the Equation

We will now add 91 to both sides of the equation to isolate the term with y.

{ 26y = -39 + 91 \}

{ 26y = 52 \}

Step 5: Divide Both Sides of the Equation by 26

We will now divide both sides of the equation by 26 to solve for y.

{ y = \frac{52}{26} \}

{ y = 2 \}

Step 6: Substitute the Value of y into the Expression for x

Now that we have the value of y, we can substitute it into the expression for x.

{ x = 3(2) - 13 \}

{ x = 6 - 13 \}

{ x = -7 \}

Conclusion

We have now solved the system of equations using the substitution method. The solution is x = -7 and y = 2.

Options

Based on the solution, we can determine the correct option.

One or more solutions: $\square$
No solution
Infinite number of solutions

The correct option is One or more solutions: $\square$.

Discussion

The substitution method is a powerful tool for solving systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This method can be used to solve systems with two or more equations and two or more variables.

In this article, we used the substitution method to solve a system of two equations with two variables. We started by solving the second equation for x and then substituted that expression into the first equation. We then simplified the equation and solved for y. Finally, we substituted the value of y into the expression for x to find the solution.

The substitution method is a useful tool for solving systems of linear equations. It can be used to solve systems with two or more equations and two or more variables. It is also a useful tool for solving systems with non-linear equations.

Example Problems

Here are some example problems that can be solved using the substitution method.

Example 1

{ \begin{cases} 2x + 3y = 7 \\ x = 2y - 1 \end{cases} \}

Solution

We will start by solving the second equation for x.

{ x = 2y - 1 \}

We will now substitute this expression into the first equation.

{ 2(2y - 1) + 3y = 7 \}

We will now simplify the equation by distributing the 2 and combining like terms.

{ 4y - 2 + 3y = 7 \}

{ 7y - 2 = 7 \}

We will now add 2 to both sides of the equation to isolate the term with y.

{ 7y = 7 + 2 \}

{ 7y = 9 \}

We will now divide both sides of the equation by 7 to solve for y.

{ y = \frac{9}{7} \}

We will now substitute the value of y into the expression for x.

{ x = 2(\frac{9}{7}) - 1 \}

{ x = \frac{18}{7} - 1 \}

{ x = \frac{18}{7} - \frac{7}{7} \}

{ x = \frac{11}{7} \}

The solution is x = 11/7 and y = 9/7.

Example 2

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

Solution

We will start by solving the second equation for x.

{ x = y + 1 \}

We will now substitute this expression into the first equation.

{ y + 1 + 2y = 3 \}

We will now simplify the equation by combining like terms.

{ 3y + 1 = 3 \}

We will now subtract 1 from both sides of the equation to isolate the term with y.

{ 3y = 3 - 1 \}

{ 3y = 2 \}

We will now divide both sides of the equation by 3 to solve for y.

{ y = \frac{2}{3} \}

We will now substitute the value of y into the expression for x.

{ x = \frac{2}{3} + 1 \}

{ x = \frac{2}{3} + \frac{3}{3} \}

{ x = \frac{5}{3} \}

The solution is x = 5/3 and y = 2/3.

Conclusion

In this article, we explored the method of substitution to solve a system of linear equations. We used the substitution method to solve a system of two equations with two variables. We started by solving the second equation for x and then substituted that expression into the first equation. We then simplified the equation and solved for y. Finally, we substituted the value of y into the expression for x to find the solution.

We also provided example problems that can be solved using the substitution method. These examples demonstrate how to use the substitution method to solve systems of linear equations.

References

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

Keywords

Substitution method
System of linear equations
Linear algebra
Algebra
Trigonometry
Calculus

Final Thoughts

We hope that this article has provided a clear understanding of the substitution method and how to use it to solve systems of linear equations.

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Introduction

In our previous article, we explored the method of substitution to solve a system of linear equations. We used the substitution method to solve a system of two equations with two variables. In this article, we will answer some frequently asked questions about solving systems of linear equations by substitution.

Q: What is the substitution method?

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

Q: When can I use the substitution method?

A: You can use the substitution method to solve systems of linear equations with two or more equations and two or more variables.

Q: How do I choose which equation to solve first?

A: You can choose which equation to solve first based on which variable is easiest to solve for. For example, if one equation has a variable with a coefficient of 1, it may be easier to solve for that variable first.

Q: What if I get a fraction or decimal in my solution?

A: If you get a fraction or decimal in your solution, you can simplify it by dividing both the numerator and denominator by their greatest common divisor (GCD) or by multiplying both the numerator and denominator by a power of 10.

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

A: Yes, you can use the substitution method to solve systems with non-linear equations. However, you may need to use additional techniques, such as factoring or the quadratic formula, to solve the equations.

Q: What if I get a system with no solution or infinite solutions?

A: If you get a system with no solution, it means that the equations are inconsistent and there is no solution. If you get a system with infinite solutions, it means that the equations are dependent and there are infinitely many solutions.

Q: Can I use the substitution method to solve systems with three or more equations?

A: Yes, you can use the substitution method to solve systems with three or more equations. However, you may need to use additional techniques, such as the elimination method or the matrix method, to solve the system.

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 it into the other equation
Not simplifying the equation after substituting the expression
Not checking for extraneous solutions
Not using the correct method for solving the equation (e.g. factoring, quadratic formula)

Q: How do I know if I have solved the system correctly?

A: To know if you have solved the system correctly, you can check your solution by plugging it back into the original equations. If the solution satisfies both equations, then you have solved the system correctly.

Q: Can I use the substitution method to solve systems with complex numbers?

A: Yes, you can use the substitution method to solve systems with complex numbers. However, you may need to use additional techniques, such as the quadratic formula or the polar form of complex numbers, to solve the equations.

Conclusion

In this article, we answered some frequently asked questions about solving systems of linear equations by substitution. We covered topics such as when to use the substitution method, how to choose which equation to solve first, and how to avoid common mistakes. We also discussed how to check if you have solved the system correctly and how to use the substitution method to solve systems with complex numbers.

References

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

Keywords

Substitution method
System of linear equations
Linear algebra
Algebra
Trigonometry
Calculus
Complex numbers

Final Thoughts

We hope that this article has provided a clear understanding of the substitution method and how to use it to solve systems of linear equations.