Solve Using Substitution:$\[ \begin{align*} x - Y &= -13 \\ -6x - 5y &= -10 \end{align*} \\]

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Introduction

In this article, we will explore the method of substitution to solve a system of linear equations. This method is a powerful tool for solving systems of equations and is often used in various fields such as mathematics, physics, and engineering. We will use a specific example to illustrate the steps involved in solving a system of linear equations using substitution.

The System of Linear Equations

The system of linear equations we will be solving is given by:

x−y=−13−6x−5y=−10\begin{align*} x - y &= -13 \\ -6x - 5y &= -10 \end{align*}

Step 1: Solve the First Equation for x

To solve the system of linear equations using substitution, we need to solve one of the equations for one of the variables. Let's solve the first equation for x.

x−y=−13x - y = -13

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

x=−13+yx = -13 + y

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

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

−6x−5y=−10-6x - 5y = -10

Substituting x = -13 + y into the second equation, we get:

−6(−13+y)−5y=−10-6(-13 + y) - 5y = -10

Step 3: Simplify the Equation

Expanding the equation, we get:

78−6y−5y=−1078 - 6y - 5y = -10

Combine like terms:

78−11y=−1078 - 11y = -10

Step 4: Solve for y

Now we can solve for y by isolating it on one side of the equation.

−11y=−88-11y = -88

Divide both sides by -11:

y=8y = 8

Step 5: Substitute the Value of y back into the Expression for x

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

x=−13+yx = -13 + y

Substituting y = 8 into the expression for x, we get:

x=−13+8x = -13 + 8

Step 6: Simplify the Expression for x

Simplifying the expression for x, we get:

x=−5x = -5

Conclusion

In this article, we used the method of substitution to solve a system of linear equations. We solved the first equation for x and then substituted the expression for x into the second equation. We then simplified the equation and solved for y. Finally, we substituted the value of y back into the expression for x to find the value of x. The solution to the system of linear equations is x = -5 and y = 8.

Example Use Cases

The method of substitution is a powerful tool for solving systems of linear equations and has many practical applications in various fields such as:

  • Physics: To solve problems involving motion, forces, and energies.
  • Engineering: To design and optimize systems, such as electrical circuits, mechanical systems, and structural systems.
  • Computer Science: To solve problems involving algorithms, data structures, and computer networks.

Tips and Tricks

When using the method of substitution to solve a system of linear equations, keep the following tips and tricks in mind:

  • Solve one equation for one variable: This will make it easier to substitute the expression into the other equation.
  • Substitute carefully: Make sure to substitute the expression for the variable into the correct equation.
  • Simplify the equation: Combine like terms and simplify the equation to make it easier to solve.

Conclusion

In conclusion, the method of substitution is a powerful tool for solving systems of linear equations. By following the steps outlined in this article, you can use the method of substitution to solve a wide range of problems involving systems of linear equations. With practice and experience, you will become proficient in using the method of substitution and be able to apply it to a variety of real-world problems.

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Q: What is the method of substitution in solving systems of linear equations?

A: The method of substitution is a technique used to solve systems of linear equations by substituting the expression for one variable into the other equation. This method is useful when one of the equations is easily solvable for one of the variables.

Q: When should I use the method of substitution to solve a system of linear equations?

A: You should use the method of substitution when one of the equations is easily solvable for one of the variables. This method is particularly useful when the coefficients of one of the variables are small or when one of the equations is a simple linear equation.

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

A: To determine which equation to solve for the variable, look for the equation that has the smallest coefficient for the variable you want to solve for. This will make it easier to substitute the expression into the other equation.

Q: What if I have a system of linear equations with three or more variables?

A: If you have a system of linear equations with three or more variables, you can use the method of substitution to solve for two of the variables and then use the method of elimination to solve for the remaining variable.

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

A: Yes, you can use the method of substitution to solve a system of linear equations with fractions. However, you will need to clear the fractions by multiplying both sides of the equation by the least common multiple of the denominators.

Q: What if I make a mistake while using the method of substitution?

A: If you make a mistake while using the method of substitution, you may end up with an incorrect solution. To avoid this, make sure to check your work by plugging the solution back into the original equations.

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

A: Yes, you can use the method of substitution to solve a system of linear equations with decimals. However, you will need to round the coefficients and constants to the correct number of decimal places to ensure accuracy.

Q: How do I know if the solution to the system of linear equations is unique?

A: To determine if the solution to the system of linear equations is unique, check if the two equations are parallel or if they intersect at a single point. If the equations are parallel, the system has no solution. If they intersect at a single point, the system has a unique solution.

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

A: Yes, you can use the method of substitution to solve a system of linear equations with absolute values. However, you will need to consider the cases where the absolute value is positive and negative separately.

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

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

  • Not checking the work by plugging the solution back into the original equations.
  • Not simplifying the equation after substitution.
  • Not considering the cases where the absolute value is positive and negative separately.
  • Not rounding the coefficients and constants to the correct number of decimal places.

Conclusion

In conclusion, the method of substitution is a powerful tool for solving systems of linear equations. By following the steps outlined in this article and avoiding common mistakes, you can use the method of substitution to solve a wide range of problems involving systems of linear equations. With practice and experience, you will become proficient in using the method of substitution and be able to apply it to a variety of real-world problems.