To Eliminate The X Terms And Solve For Y In The Fewest Steps, By Which Constants Should The Equations Be Multiplied Before Adding The Equations Together?First Equation: 6 X − 5 Y = 17 6x - 5y = 17 6 X − 5 Y = 17 Second Equation: 7 X + 3 Y = 11 7x + 3y = 11 7 X + 3 Y = 11 A. The First

by ADMIN 285 views

Multiplying Equations to Eliminate Variables

When solving a system of linear equations, it is often necessary to eliminate one or more variables to find the solution. In this case, we are given two equations with two variables, x and y. Our goal is to eliminate the x terms and solve for y in the fewest steps possible. To do this, we need to multiply the equations by suitable constants before adding them together.

The First Equation

The first equation is:

6x−5y=176x - 5y = 17

The Second Equation

The second equation is:

7x+3y=117x + 3y = 11

Multiplying the Equations

To eliminate the x terms, we need to multiply the equations by constants that will make the coefficients of x in both equations opposite. In other words, we need to multiply the first equation by a constant that will make the coefficient of x in the first equation equal to the negative of the coefficient of x in the second equation.

Let's analyze the coefficients of x in both equations:

  • In the first equation, the coefficient of x is 6.
  • In the second equation, the coefficient of x is 7.

To make the coefficients of x opposite, we need to multiply the first equation by a constant that will make the coefficient of x in the first equation equal to -7. Since the coefficient of x in the first equation is 6, we need to multiply the first equation by -7/6.

Similarly, we need to multiply the second equation by a constant that will make the coefficient of x in the second equation equal to 6. Since the coefficient of x in the second equation is 7, we need to multiply the second equation by -6/7.

Multiplying the First Equation

Let's multiply the first equation by -7/6:

−76(6x−5y)=−76(17)\frac{-7}{6}(6x - 5y) = \frac{-7}{6}(17)

Simplifying the equation, we get:

−7x+356y=−1196-7x + \frac{35}{6}y = -\frac{119}{6}

Multiplying the Second Equation

Let's multiply the second equation by -6/7:

−67(7x+3y)=−67(11)\frac{-6}{7}(7x + 3y) = \frac{-6}{7}(11)

Simplifying the equation, we get:

−427x−187y=−667-\frac{42}{7}x - \frac{18}{7}y = -\frac{66}{7}

Adding the Equations

Now that we have multiplied the equations by suitable constants, we can add them together to eliminate the x terms:

−7x+356y=−1196-7x + \frac{35}{6}y = -\frac{119}{6}

−427x−187y=−667-\frac{42}{7}x - \frac{18}{7}y = -\frac{66}{7}

Adding the two equations, we get:

356y−187y=−1196−667\frac{35}{6}y - \frac{18}{7}y = -\frac{119}{6} - \frac{66}{7}

Simplifying the equation, we get:

24542y−10842y=−1196−667\frac{245}{42}y - \frac{108}{42}y = -\frac{119}{6} - \frac{66}{7}

Combining like terms, we get:

13742y=−1196−667\frac{137}{42}y = -\frac{119}{6} - \frac{66}{7}

Simplifying the Right-Hand Side

To simplify the right-hand side, we need to find a common denominator for the fractions. The least common multiple of 6 and 7 is 42. Therefore, we can rewrite the fractions as:

−1196=−119×76×7=−83342-\frac{119}{6} = -\frac{119 \times 7}{6 \times 7} = -\frac{833}{42}

−667=−66×67×6=−39642-\frac{66}{7} = -\frac{66 \times 6}{7 \times 6} = -\frac{396}{42}

Now we can add the fractions:

−83342−39642=−122942-\frac{833}{42} - \frac{396}{42} = -\frac{1229}{42}

Simplifying the Equation

Now that we have simplified the right-hand side, we can rewrite the equation as:

13742y=−122942\frac{137}{42}y = -\frac{1229}{42}

To solve for y, we can multiply both sides of the equation by the reciprocal of the coefficient of y:

y=−122942×42137y = -\frac{1229}{42} \times \frac{42}{137}

Simplifying the equation, we get:

y=−1229137y = -\frac{1229}{137}

Therefore, the value of y is -9.

Conclusion

In this article, we have shown how to eliminate the x terms and solve for y in the fewest steps possible by multiplying the equations by suitable constants before adding them together. We have used the first equation:

6x−5y=176x - 5y = 17

and the second equation:

7x+3y=117x + 3y = 11

Q: What is the purpose of multiplying equations by constants before adding them together?

A: The purpose of multiplying equations by constants before adding them together is to eliminate one or more variables. By multiplying the equations by suitable constants, we can make the coefficients of the variables opposite, allowing us to add the equations together and eliminate the variables.

Q: How do I determine the constants to multiply the equations by?

A: To determine the constants to multiply the equations by, you need to analyze the coefficients of the variables in both equations. You need to find the constants that will make the coefficients of the variables opposite. This can be done by dividing the coefficient of one variable by the coefficient of the same variable in the other equation.

Q: What if the coefficients of the variables are not integers?

A: If the coefficients of the variables are not integers, you can still multiply the equations by the necessary constants. However, you may need to simplify the resulting equation to eliminate any fractions.

Q: Can I use this method to solve systems of linear equations with more than two variables?

A: Yes, you can use this method to solve systems of linear equations with more than two variables. However, you will need to eliminate one variable at a time, using the method described above.

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

A: If you have a system of linear equations with no solution, it means that the equations are inconsistent. In this case, you will not be able to eliminate the variables using the method described above.

Q: Can I use this method to solve systems of linear equations with dependent variables?

A: Yes, you can use this method to solve systems of linear equations with dependent variables. However, you will need to eliminate one variable at a time, using the method described above.

Q: What are some common mistakes to avoid when using this method?

A: Some common mistakes to avoid when using this method include:

  • Not analyzing the coefficients of the variables correctly
  • Not multiplying the equations by the correct constants
  • Not simplifying the resulting equation correctly
  • Not checking for inconsistencies in the system of equations

Q: Can I use this method to solve systems of linear equations with complex coefficients?

A: Yes, you can use this method to solve systems of linear equations with complex coefficients. However, you will need to use complex arithmetic to simplify the resulting equation.

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

A: This method has many real-world applications, including:

  • Solving systems of linear equations in physics and engineering
  • Solving systems of linear equations in economics and finance
  • Solving systems of linear equations in computer science and data analysis

Q: Can I use this method to solve systems of linear equations with multiple solutions?

A: Yes, you can use this method to solve systems of linear equations with multiple solutions. However, you will need to use the method described above to eliminate one variable at a time, and then use the resulting equation to find the multiple solutions.

Q: What are some tips for using this method effectively?

A: Some tips for using this method effectively include:

  • Carefully analyzing the coefficients of the variables
  • Multiplying the equations by the correct constants
  • Simplifying the resulting equation correctly
  • Checking for inconsistencies in the system of equations
  • Using complex arithmetic when necessary

By following these tips and avoiding common mistakes, you can use this method to solve systems of linear equations effectively and efficiently.