Figure Out What To Multiply The Second Equation By To Help You Eliminate The $x$'s Or The $y$'s.$\[ \begin{array}{c} 10x - 9y = -5 \\ -2x + 3y = 10 \end{array} \\]

Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. There are several methods to solve a system of linear equations, including the substitution method, the elimination method, and the graphing method. In this article, we will focus on the elimination method, which involves eliminating one of the variables by multiplying the equations by appropriate constants.

The Elimination Method

The elimination method involves multiplying the equations by appropriate constants to eliminate one of the variables. To do this, we need to figure out what to multiply the second equation by to help us eliminate the x's or the y's. Let's consider the following system of linear equations:

$$\begin{array}{c} 10x - 9y = -5 \\ -2x + 3y = 10 \end{array}$$

Step 1: Identify the Coefficients

To eliminate one of the variables, we need to identify the coefficients of the variables in both equations. The coefficients are the numbers that are multiplied by the variables. In the first equation, the coefficient of x is 10 and the coefficient of y is -9. In the second equation, the coefficient of x is -2 and the coefficient of y is 3.

Step 2: Determine the Multiplication Factors

To eliminate one of the variables, we need to determine the multiplication factors that will make the coefficients of the variables in both equations equal. Let's say we want to eliminate the x's. We can multiply the first equation by 2 and the second equation by 5 to make the coefficients of x equal.

Step 3: Multiply the Equations

Now that we have determined the multiplication factors, we can multiply the equations by these factors.

$$\begin{array}{c} (10x - 9y) \times 2 = (-5) \times 2 \\ (-2x + 3y) \times 5 = 10 \times 5 \end{array}$$

This gives us:

$$\begin{array}{c} 20x - 18y = -10 \\ -10x + 15y = 50 \end{array}$$

Step 4: Add the Equations

Now that we have multiplied the equations, we can add them to eliminate the x's.

$$\begin{array}{c} (20x - 18y) + (-10x + 15y) = (-10) + 50 \\ 10x - 3y = 40 \end{array}$$

Step 5: Solve for y

Now that we have eliminated the x's, we can solve for y.

$$-3y = 40$$

Dividing both sides by -3 gives us:

$$y = -\frac{40}{3}$$

Step 6: Substitute y into One of the Original Equations

Now that we have found the value of y, we can substitute it into one of the original equations to find the value of x.

Let's substitute y into the first equation:

$$10x - 9y = -5$$

Substituting y = -40/3 gives us:

$$10x - 9(-\frac{40}{3}) = -5$$

Simplifying this equation gives us:

$$10x + \frac{360}{3} = -5$$

$$10x + 120 = -5$$

Subtracting 120 from both sides gives us:

$$10x = -125$$

Dividing both sides by 10 gives us:

$$x = -\frac{125}{10}$$

$$x = -\frac{25}{2}$$

Conclusion

In this article, we have used the elimination method to solve a system of linear equations. We have identified the coefficients of the variables, determined the multiplication factors, multiplied the equations, added the equations, solved for y, and substituted y into one of the original equations to find the value of x. The elimination method is a powerful tool for solving systems of linear equations, and it can be used to solve systems with two or more variables.

Example Problems

Here are some example problems that you can try to practice the elimination method:

1. Solve the following system of linear equations using the elimination method:

$$\begin{array}{c} 2x + 3y = 7 \\ x - 2y = -3 \end{array}$$

2. Solve the following system of linear equations using the elimination method:

$$\begin{array}{c} x + 2y = 4 \\ 3x - 2y = 5 \end{array}$$

3. Solve the following system of linear equations using the elimination method:

$$\begin{array}{c} x - 3y = -2 \\ 2x + y = 3 \end{array}$$

Tips and Tricks

Here are some tips and tricks that you can use to help you solve systems of linear equations using the elimination method:

1. Make sure to identify the coefficients of the variables in both equations.
2. Determine the multiplication factors that will make the coefficients of the variables in both equations equal.
3. Multiply the equations by the multiplication factors.
4. Add the equations to eliminate one of the variables.
5. Solve for the variable that is not eliminated.
6. Substitute the value of the eliminated variable into one of the original equations to find the value of the other variable.

Q: What is the elimination method?

A: The elimination method is a technique used to solve systems of linear equations by eliminating one of the variables. This is done by multiplying the equations by appropriate constants to make the coefficients of the variables equal, and then adding the equations to eliminate the variable.

Q: How do I know which variable to eliminate?

A: You can eliminate either variable, but it's often easier to eliminate the variable that has the smaller coefficient. This will make the calculations simpler and reduce the chance of errors.

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

A: If the coefficients of the variables are not equal, you can multiply the equations by appropriate constants to make them equal. This is done by finding the least common multiple (LCM) of the coefficients and multiplying both equations by this value.

Q: Can I use the elimination method with systems of linear equations with more than two variables?

A: Yes, you can use the elimination method with systems of linear equations with more than two variables. However, it's often more complicated and may require more steps.

Q: What if I get a contradiction when I add the equations?

A: If you get a contradiction when you add the equations, it means that the system of linear equations has no solution. This can happen if the equations are inconsistent or if there is a mistake in the calculations.

Q: Can I use the elimination method with systems of linear equations with fractions?

A: Yes, you can use the elimination method with systems of linear equations with fractions. However, you may need to multiply the equations by a common denominator to eliminate the fractions.

Q: What if I get a solution that is not a whole number?

A: If you get a solution that is not a whole number, it means that the solution is a decimal or a fraction. This is a valid solution, and you should not round it to the nearest whole number.

Q: Can I use the elimination method with systems of linear equations with variables on both sides of the equation?

A: No, you cannot use the elimination method with systems of linear equations with variables on both sides of the equation. This is because the elimination method requires that the variables be isolated on one side of the equation.

Q: What if I get a solution that is not unique?

A: If you get a solution that is not unique, it means that there are multiple solutions to the system of linear equations. This can happen if the equations are dependent or if there is a mistake in the calculations.

Q: Can I use the elimination method with systems of linear equations with absolute values?

A: No, you cannot use the elimination method with systems of linear equations with absolute values. This is because the elimination method requires that the equations be linear, and absolute values are not linear.

Q: What if I get a solution that is not a real number?

A: If you get a solution that is not a real number, it means that the solution is complex or imaginary. This is a valid solution, and you should not ignore it.

Conclusion

The elimination method is a powerful tool for solving systems of linear equations. By following the steps outlined in this article, you can use the elimination method to solve systems of linear equations with ease. Remember to identify the coefficients of the variables, determine the multiplication factors, multiply the equations, add the equations, and solve for the variables. With practice, you will become proficient in using the elimination method to solve systems of linear equations.

Example Problems

Here are some example problems that you can try to practice the elimination method:

1. Solve the following system of linear equations using the elimination method:

$$\begin{array}{c} 2x + 3y = 7 \\ x - 2y = -3 \end{array}$$

2. Solve the following system of linear equations using the elimination method:

$$\begin{array}{c} x + 2y = 4 \\ 3x - 2y = 5 \end{array}$$

3. Solve the following system of linear equations using the elimination method:

$$\begin{array}{c} x - 3y = -2 \\ 2x + y = 3 \end{array}$$

Tips and Tricks

Here are some tips and tricks that you can use to help you solve systems of linear equations using the elimination method:

1. Make sure to identify the coefficients of the variables in both equations.
2. Determine the multiplication factors that will make the coefficients of the variables equal.
3. Multiply the equations by the multiplication factors.
4. Add the equations to eliminate one of the variables.
5. Solve for the variable that is not eliminated.
6. Substitute the value of the eliminated variable into one of the original equations to find the value of the other variable.

By following these tips and tricks, you can use the elimination method to solve systems of linear equations with ease.