Solving Linear Equations: A Step-by-Step Guide

Hey guys! Let's dive into solving a system of linear equations. It's like a puzzle, and we're going to break it down step by step to make it super easy to understand. We're going to use the elimination method, which is a super useful technique in algebra. The system of equations we are solving is:

$\begin{array}{l} 3 x-y=-18 \\ 4 x+3 y=-11 \\ \end{array}$

Now, solving systems of linear equations is a fundamental concept in mathematics, used extensively in various fields, from science and engineering to economics and computer graphics. It helps us find points where multiple lines intersect, which can represent solutions to real-world problems. By learning how to solve these equations, you will gain a valuable skill that will help you in many other math topics. So, let’s get started.

Step 1: Prepare the Equations for Elimination

The goal of the elimination method is to eliminate one of the variables (either x or y) by adding or subtracting the equations. To do this effectively, we need to manipulate the equations so that the coefficients of one of the variables are opposites. Looking at our equations:

$\begin{array}{l} 3 x-y=-18 \\ 4 x+3 y=-11 \\ \end{array}$

Notice that the coefficients of y are -1 and 3. If we multiply the first equation by 3, the coefficients of y will become -3 and 3, which are opposites. So, let's do that! Multiplying the first equation by 3 gives us:

$3 * (3x - y) = 3 * (-18)$

This simplifies to:

$9x - 3y = -54$

Now, our system of equations looks like this:

$\begin{array}{l} 9 x-3 y=-54 \\ 4 x+3 y=-11 \\ \end{array}$

We haven't changed the fundamental relationship between x and y; we've just rewritten the first equation in a way that makes it easier to eliminate y. Remember, the key is to ensure we are doing the same operation to both sides of the equation to maintain balance.

Now, we are ready to proceed with the next step, which is adding the equations together. This will eliminate the variable y and allow us to solve for x. The preparation step is essential, as it sets the stage for the actual elimination, simplifying the process and making it more efficient.

Why is preparing the equations important?

• Simplifies the process: Preparing equations allows us to eliminate a variable directly, simplifying calculations. Imagine trying to solve for x or y without getting rid of one of them first; it'd be much more complex. This preparation is a crucial organizational step. It’s like setting up the pieces on a chessboard before starting the game.
• Reduces errors: By carefully multiplying the equations, we create opportunities for cancellation, which minimizes errors. It helps us keep the arithmetic clean and clear.
• Efficiency: Preparing the equations helps solve the system of equations. Without this, we would have to employ much more complex methods.

Step 2: Eliminate One Variable

Now that we have prepared our equations, we're ready to eliminate one of the variables. We will add the two equations together. Here are our equations again:

$\begin{array}{l} 9 x-3 y=-54 \\ 4 x+3 y=-11 \\ \end{array}$

Add the equations, term by term:

$(9x + 4x) + (-3y + 3y) = (-54 + -11)$

This simplifies to:

$13x + 0y = -65$

Which further simplifies to:

$13x = -65$

See how the y variable is gone? That's the power of the elimination method! This step is all about isolating one variable, making it easier to solve for its value. The preparation step was crucial, and now we can continue to the next step. By eliminating one variable, we have transformed the system of two equations into a single equation with only one variable, which is much easier to solve.

The Importance of Correct Addition and Subtraction

• Attention to Signs: When adding or subtracting equations, paying close attention to the signs (+ or -) of the terms is extremely important. A small mistake in the sign can completely change the result. For example, in the previous step, we added $-3y + 3y$ and got $0y$, which is correct because the coefficients of y were opposite signs.
• Term Alignment: Make sure you're adding or subtracting the correct terms. Adding x terms to y terms (or vice versa) can mess up the whole process. Ensure that you’re adding the x term with the x term, the y term with the y term, and the constants with constants.
• Double-check: After adding or subtracting, always double-check your work to avoid making calculation errors. Even small calculation mistakes can cause significant errors in the final solution. Take a moment to review the math.

Step 3: Solve for the Remaining Variable

Now we've got a much simpler equation to deal with:

$13x = -65$

To solve for x, we need to isolate it by dividing both sides of the equation by 13:

$x = -65 / 13$

$x = -5$

And there we have it! We've found the value of x. This step is straightforward, but it's crucial to correctly perform the division or any other required operation to isolate the variable. The process of solving a single-variable equation is a skill used many times in mathematics. This is why we have to learn it very well.

Tips for Solving Single-Variable Equations

• Simplify first: Before solving, try to simplify the equation as much as possible. This may involve combining like terms or performing any preliminary operations. This will make your calculations easier and reduce the chance of errors. For example, if you see terms that can be added or subtracted, do it first.
• Isolate the variable: To isolate a variable, perform operations on both sides of the equation. Ensure you perform the same operation on both sides of the equation to maintain balance and accuracy. Whatever you do to one side of the equation, you must do to the other to keep the equation valid.
• Check your work: Always check your answer by plugging it back into the original equation. This is the surest way to confirm that your solution is correct. To do this, substitute the value of x back into the original equation and solve to see if it holds true.

Step 4: Substitute to Find the Other Variable

We know that x = -5. Now we need to find the value of y. We can do this by substituting the value of x into either of the original equations. Let's use the first equation:

$3x - y = -18$

Substitute x = -5:

$3 * (-5) - y = -18$

This simplifies to:

$-15 - y = -18$

Now, we need to solve for y. Add 15 to both sides:

$-y = -18 + 15$

$-y = -3$

Finally, multiply both sides by -1 to solve for y:

$y = 3$

So, y = 3. Now we have both the x and y values, which means we have solved the system of equations. This is a very important step. Remember to carefully substitute the value you found for the first variable into one of the original equations. The correct substitution ensures that the solution meets all the conditions of the original equations.

Avoiding Mistakes in Substitution

• Careful Substitution: The most common mistake here is incorrectly substituting the value of x into the equation. Double-check that you're using the correct equation and that you're substituting the x value in the correct place.
• Simplify Carefully: After substituting, simplify the equation step by step. This minimizes the risk of making arithmetic errors. Follow the order of operations (PEMDAS/BODMAS) to ensure you perform the operations in the correct sequence.
• Verify the result: Always verify your answer by substituting both the x and y values into both original equations. If both equations are true, you know your answer is correct. This is like checking your work to make sure you didn’t miss anything.

Step 5: State the Solution

We found x = -5 and y = 3. We can write the solution as an ordered pair (x, y). Therefore, the solution to the system of equations is:

(-5, 3)

This means that the point (-5, 3) is the point where the two lines represented by the equations intersect. This step is about clearly presenting your results. This format is standard and easy to understand. Stating the solution clearly is the final and crucial step. By clearly stating the solution, you ensure that the solution can be interpreted correctly. The solution indicates the point on the coordinate plane where the two lines intersect, satisfying the conditions of both original equations.

Presenting Your Solution Clearly

• Use Ordered Pairs: Always present your solution as an ordered pair (x, y). This format is standard and easy to understand. In the coordinate plane, the order of the numbers is important to know the exact position of the lines.
• Double-Check the Order: Ensure you have the values of x and y in the correct order. The x-value always comes first, followed by the y-value.
• Verify Your Answer: Before stating the solution, verify your work once more. Substitute your x and y values back into the original equations to make sure both equations are true. This final check can help you catch any small errors you might have made along the way.

Conclusion

Awesome, guys! We've successfully solved the system of linear equations using the elimination method. We went through it step-by-step: preparing the equations, eliminating a variable, solving for the remaining variable, substituting to find the other variable, and stating the solution. Remember, practice makes perfect! The more you practice, the easier it will become. You are building essential skills that will benefit you in future math topics. Keep up the great work! You have now mastered a fundamental skill in algebra.