Malik Used These Steps To Solve The System Of Equations:${ \begin{align*} 2x + 4y &= 20 \ 5x - 3y &= 11 \end{align*} }$[ \begin{array}{|l|c|} \hline \text{Step 1} & \begin{array}{c} 6x + 12y = 60 \ 20x - 12y = 44 \end{array}

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. A system of linear equations is a set of two or more linear equations that involve two or more variables. In this article, we will focus on solving a system of two linear equations with two variables using the method of elimination.

The Method of Elimination

The method of elimination is a popular technique used to solve systems of linear equations. This method involves adding or subtracting the equations in such a way that one of the variables is eliminated. The goal is to eliminate one of the variables by making the coefficients of that variable the same in both equations, but with opposite signs.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to make the coefficients of that variable the same in both equations, but with opposite signs. We can do this by multiplying the equations by necessary multiples.

Let's consider the following system of linear equations:

{ \begin{align*} 2x + 4y &= 20 \\ 5x - 3y &= 11 \end{align*} \}

To eliminate the variable $y$, we can multiply the first equation by 3 and the second equation by 4.

{ \begin{array}{|l|c|} \hline \text{Step 1} & \begin{array}{c} 6x + 12y = 60 \\ 20x - 12y = 44 \end{array} \end{array} \}

Step 2: Add or Subtract the Equations

Now that we have the equations with the coefficients of $y$ being the same but with opposite signs, we can add or subtract the equations to eliminate the variable $y$.

Let's add the two equations:

{ \begin{array}{|l|c|} \hline \text{Step 2} & \begin{array}{c} (6x + 12y) + (20x - 12y) = 60 + 44 \\ 26x = 104 \end{array} \end{array} \}

Step 3: Solve for the Variable

Now that we have the equation with only one variable, we can solve for that variable.

Let's solve for $x$:

{ \begin{array}{|l|c|} \hline \text{Step 3} & \begin{array}{c} 26x = 104 \\ x = \frac{104}{26} \\ x = 4 \end{array} \end{array} \}

Step 4: Substitute the Value of the Variable

Now that we have the value of one variable, we can substitute it into one of the original equations to solve for the other variable.

Let's substitute the value of $x$ into the first equation:

{ \begin{array}{|l|c|} \hline \text{Step 4} & \begin{array}{c} 2(4) + 4y = 20 \\ 8 + 4y = 20 \\ 4y = 12 \\ y = \frac{12}{4} \\ y = 3 \end{array} \end{array} \}

Conclusion

In this article, we have solved a system of two linear equations with two variables using the method of elimination. We have shown that by multiplying the equations by necessary multiples, adding or subtracting the equations, solving for the variable, and substituting the value of the variable, we can find the solution to the system of equations.

Real-World Applications

Solving systems of linear equations has numerous real-world applications. For example, in physics, we can use systems of linear equations to model the motion of objects. In engineering, we can use systems of linear equations to design and optimize systems. In economics, we can use systems of linear equations to model the behavior of markets.

Tips and Tricks

Here are some tips and tricks to help you solve systems of linear equations:

• Make sure to read the problem carefully and understand what is being asked.
• Use the method of elimination to eliminate one of the variables.
• Multiply the equations by necessary multiples to make the coefficients of the variable the same but with opposite signs.
• Add or subtract the equations to eliminate the variable.
• Solve for the variable and substitute the value of the variable into one of the original equations to solve for the other variable.

Common Mistakes

Here are some common mistakes to avoid when solving systems of linear equations:

• Not reading the problem carefully and understanding what is being asked.
• Not using the method of elimination to eliminate one of the variables.
• Not multiplying the equations by necessary multiples to make the coefficients of the variable the same but with opposite signs.
• Not adding or subtracting the equations to eliminate the variable.
• Not solving for the variable and substituting the value of the variable into one of the original equations to solve for the other variable.

Conclusion

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields. In our previous article, we provided a step-by-step guide on how to solve systems of linear equations using the method of elimination. In this article, we will answer some of the most frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that involve two or more variables. For example:

{ \begin{align*} 2x + 4y &= 20 \\ 5x - 3y &= 11 \end{align*} \}

Q: What is the method of elimination?

The method of elimination is a popular technique used to solve systems of linear equations. This method involves adding or subtracting the equations in such a way that one of the variables is eliminated. The goal is to eliminate one of the variables by making the coefficients of that variable the same in both equations, but with opposite signs.

Q: How do I know which variable to eliminate first?

To determine which variable to eliminate first, you need to look at the coefficients of the variables in both equations. If the coefficients of one variable are the same but with opposite signs, you can eliminate that variable first. If not, you need to multiply one or both equations by necessary multiples to make the coefficients of one variable the same but with opposite signs.

Q: What is the difference between adding and subtracting equations?

When adding equations, you are combining the two equations by adding the corresponding terms. When subtracting equations, you are combining the two equations by subtracting the corresponding terms. For example:

{ \begin{array}{|l|c|} \hline \text{Adding equations} & \begin{array}{c} (2x + 4y) + (5x - 3y) = 20 + 11 \\ 7x + y = 31 \end{array} \end{array} \}

{ \begin{array}{|l|c|} \hline \text{Subtracting equations} & \begin{array}{c} (2x + 4y) - (5x - 3y) = 20 - 11 \\ -3x + 7y = 9 \end{array} \end{array} \}

Q: How do I solve for the variable after eliminating it?

After eliminating one of the variables, you can solve for the other variable by isolating it on one side of the equation. For example:

{ \begin{array}{|l|c|} \hline \text{Solving for x} & \begin{array}{c} 7x + y = 31 \\ 7x = 31 - y \\ x = \frac{31 - y}{7} \end{array} \end{array} \}

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

If you have a system of linear equations with three or more variables, you can use the method of elimination to solve for two variables, and then substitute the values of those variables into one of the original equations to solve for the third variable.

Q: Can I use other methods to solve systems of linear equations?

Yes, there are other methods to solve systems of linear equations, such as the method of substitution and the method of matrices. However, the method of elimination is one of the most popular and widely used methods.

Q: What are some common mistakes to avoid when solving systems of linear equations?

Some common mistakes to avoid when solving systems of linear equations include:

• Not reading the problem carefully and understanding what is being asked.
• Not using the method of elimination to eliminate one of the variables.
• Not multiplying the equations by necessary multiples to make the coefficients of the variable the same but with opposite signs.
• Not adding or subtracting the equations to eliminate the variable.
• Not solving for the variable and substituting the value of the variable into one of the original equations to solve for the other variable.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields. By following the steps outlined in this article, you can solve systems of linear equations using the method of elimination. Remember to read the problem carefully, use the method of elimination, multiply the equations by necessary multiples, add or subtract the equations, solve for the variable, and substitute the value of the variable into one of the original equations to solve for the other variable.