Solving Systems Of Equations By Substitution A Step By Step Guide

Hey guys! Today, we're diving into a common yet crucial topic in algebra: solving systems of equations using the substitution method. Specifically, we're going to tackle the following system:

$$\left\{ \begin{array}{l} 6x - 4y = 30 \\ x = -7y - 18 \end{array} \right.$$

We'll break down each step, making sure you understand not just how to solve it, but why each step is necessary. By the end of this guide, you'll be a substitution pro! Let's get started!

Understanding the Substitution Method

Before we jump into the specific problem, let's quickly recap the substitution method itself. At its heart, this method is about simplifying a system of equations by expressing one variable in terms of the other. Think of it like replacing one ingredient in a recipe with an equivalent substitute. The goal is to reduce the system to a single equation with a single variable, which is much easier to solve.

In more detail, the substitution method involves these key steps:

1. Isolate one variable in one of the equations. This means getting one variable by itself on one side of the equation. We look for equations where a variable already has a coefficient of 1 or -1, as this makes the isolation process much simpler.
2. Substitute the expression for that variable into the other equation. This is the heart of the method! We're replacing the isolated variable in the second equation with the expression we found in the first step. This eliminates one variable, leaving us with an equation in only one variable.
3. Solve the resulting equation for the remaining variable. This is typically a straightforward algebraic process, involving combining like terms and isolating the variable.
4. Substitute back the value found in step 3 into either of the original equations to solve for the other variable. This gives us the value of the second variable, completing our solution.
5. Check your solution by substituting both values back into the original equations. This is a crucial step to ensure that your solution is correct.

Why does this method work? It works because we're using equivalent expressions. When we isolate a variable, we're finding an expression that is always equal to that variable. So, substituting that expression into another equation doesn't change the solution of the system. It just rewrites the system in a more manageable form.

Now, let's see how this plays out with our example problem!

Step-by-Step Solution of the System

Step 1: Identify the Easiest Variable to Isolate

Looking at our system:

$$\left\{ \begin{array}{l} 6x - 4y = 30 \\ x = -7y - 18 \end{array} \right.$$

We immediately notice that the second equation, x = -7y - 18, is already solved for x! This is fantastic news because it means we can skip the isolation step and move straight to substitution. In fact, this is one of the main advantages of the substitution method: it simplifies the process when one variable is already isolated, or easily isolated.

If neither equation had a variable already isolated, we would need to choose one to isolate. The best approach is to choose the variable with the smallest coefficient (in absolute value) and preferably a coefficient of 1 or -1. This minimizes the chances of dealing with fractions, which can make the algebra a bit messier. However, in this case, we've been given a head start, which makes the problem much easier.

Step 2: Substitute the Expression into the Other Equation

Since we know that x = -7y - 18, we can substitute this expression for x into the first equation. This is the core of the substitution method: replacing one variable with its equivalent expression in terms of the other variable.

Our first equation is 6x - 4y = 30. Replacing x with (-7y - 18), we get:

6(-7y - 18) - 4y = 30

Notice how we've replaced x with the entire expression (-7y - 18). It's crucial to use parentheses here to ensure that the 6 is distributed correctly to both terms inside the expression. This step is a game-changer because we've transformed our system of two equations into a single equation with only one variable, y. This single equation is now solvable using standard algebraic techniques.

Step 3: Solve for the Remaining Variable

Now we have the equation: 6(-7y - 18) - 4y = 30. Let's solve for y.

First, we distribute the 6:

-42y - 108 - 4y = 30

Next, we combine like terms (the y terms):

-46y - 108 = 30

Now, we isolate the y term by adding 108 to both sides:

-46y = 138

Finally, we solve for y by dividing both sides by -46:

y = -3

So, we've found that y = -3. This is a significant step forward! We now know the value of one of our variables. Now, we just need to find the value of x.

Step 4: Substitute Back to Find the Other Variable

Now that we know y = -3, we can substitute this value back into either of the original equations to solve for x. However, since we already have an equation solved for x, namely x = -7y - 18, it makes sense to use this one. This will minimize our work and reduce the chances of errors.

Substituting y = -3 into x = -7y - 18, we get:

x = -7(-3) - 18

Simplifying, we have:

x = 21 - 18

x = 3

Therefore, we've found that x = 3. We now have a potential solution: x = 3 and y = -3. But before we celebrate, we need to make sure this solution actually works!

Step 5: Check the Solution

It's essential to check our solution by substituting x = 3 and y = -3 into both of the original equations. This ensures that our solution satisfies the entire system, not just one equation. If it doesn't check out in both equations, we know we've made a mistake somewhere and need to go back and review our steps.

Let's start with the first equation, 6x - 4y = 30. Substituting our values, we get:

6(3) - 4(-3) = 30

18 + 12 = 30

30 = 30

This equation checks out! Now, let's check the second equation, x = -7y - 18:

3 = -7(-3) - 18

3 = 21 - 18

3 = 3

This equation also checks out! Since our solution satisfies both equations, we can confidently say that the solution to the system is x = 3 and y = -3.

The Solution and Its Interpretation

We have successfully solved the system using the substitution method! Our solution is x = 3 and y = -3. We can write this as an ordered pair (3, -3), which represents the point where the two lines represented by the equations intersect on a graph.

One or More Solutions?

In this case, we found a unique solution. This means the two lines intersect at exactly one point. So, the correct answer is:

One or more solutions: ✅

No Solution or Infinite Solutions?

It's worth noting that not all systems of equations have a unique solution. There are two other possibilities:

• No solution: This occurs when the lines are parallel and never intersect. In this case, when you try to solve the system, you'll end up with a contradiction, like 0 = 5. The variables will cancel out, and you'll be left with a false statement.
• Infinite number of solutions: This occurs when the lines are the same line. In this case, when you try to solve the system, you'll end up with an identity, like 0 = 0. The variables will cancel out, and you'll be left with a true statement.

In our case, since we found a unique solution (3, -3), we know that the lines intersect at a single point and are not parallel or the same line.

Conclusion: Mastering the Substitution Method

Congratulations! You've successfully navigated the substitution method and solved a system of equations. Remember, the key is to break the problem down into manageable steps: isolate a variable, substitute, solve, substitute back, and check. By following these steps carefully, you can confidently tackle any system of equations that comes your way.

The substitution method is a powerful tool in algebra, and mastering it will open doors to solving more complex problems in mathematics and other fields. Keep practicing, and you'll become a pro in no time!

If you found this guide helpful, please let me know in the comments! And if you have any other math problems you'd like me to tackle, feel free to suggest them. Keep learning, guys!