Solving Systems Of Equations By Substitution Method

by Alex Johnson 52 views

In the realm of mathematics, solving systems of equations is a fundamental skill. Among various methods, the substitution method stands out as a versatile technique. This guide will walk you through the process of solving a system of equations using substitution, focusing on clarity and step-by-step understanding. We will address the system:

-3x - 7y = -31
y = -5x - 23

and determine whether it has one or more solutions, no solution, or an infinite number of solutions.

Understanding the Substitution Method

The substitution method is a powerful algebraic technique used to solve systems of equations. A system of equations is a set of two or more equations that share variables. The goal is to find the values of the variables that satisfy all equations in the system simultaneously. The beauty of the substitution method lies in its straightforward approach: we solve one equation for one variable and then substitute that expression into the other equation. This reduces the system to a single equation with one variable, which we can then solve. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.

The method's effectiveness stems from its ability to simplify complex systems into manageable steps. It's particularly useful when one of the equations is already solved for one variable, or when it's easy to isolate a variable. However, understanding the underlying principles is crucial. The substitution method relies on the fundamental idea that if two expressions are equal, one can be substituted for the other without changing the equation's validity. This allows us to reduce the complexity of the system systematically, ultimately leading to the solution.

Step-by-Step Solution

Let's dive into solving the given system of equations:

-3x - 7y = -31  (Equation 1)
y = -5x - 23   (Equation 2)

Step 1: Identify the Variable to Substitute

Looking at our equations, Equation 2 (y = -5x - 23) is already solved for y. This makes it an ideal candidate for substitution. We know that y is equivalent to the expression -5x - 23. This is a crucial observation because it allows us to replace every instance of y in the other equation with this expression. By doing so, we effectively reduce the number of variables in Equation 1, making it solvable for x.

Step 2: Substitute the Expression

Now, substitute the expression for y from Equation 2 into Equation 1:

-3x - 7(-5x - 23) = -31

This is the core of the substitution method. We've taken the expression that defines y in terms of x and plugged it into the other equation. This eliminates y from Equation 1, leaving us with an equation that only involves x. The new equation maintains the equality of the original system, but it's simpler to solve because it has only one unknown variable.

Step 3: Simplify and Solve for x

Next, we simplify the equation and solve for x:

-3x + 35x + 161 = -31
32x + 161 = -31
32x = -192
x = -6

This step involves basic algebraic manipulations. First, we distribute the -7 across the parentheses, remembering to multiply it with both terms inside. This gives us -3x + 35x + 161 = -31. Next, we combine like terms on the left side, resulting in 32x + 161 = -31. To isolate the term with x, we subtract 161 from both sides, which gives us 32x = -192. Finally, we divide both sides by 32 to solve for x, and we find that x = -6.

Step 4: Substitute x Back to Find y

Now that we have the value of x, we can substitute it back into either Equation 1 or Equation 2 to find the value of y. Equation 2 is simpler, so we'll use that:

y = -5(-6) - 23
y = 30 - 23
y = 7

We substitute x = -6 into the equation y = -5x - 23. This gives us y = -5(-6) - 23. Multiplying -5 by -6 gives us 30, so the equation becomes y = 30 - 23. Subtracting 23 from 30 gives us the value of y, which is 7.

Step 5: State the Solution

The solution to the system of equations is the ordered pair (x, y) = (-6, 7). This means that the point (-6, 7) is the intersection of the two lines represented by the equations in the system. It's the one point that satisfies both equations simultaneously.

Determining the Number of Solutions

In this case, we found a unique solution for x and y. This indicates that the system has one solution. Graphically, this means the two lines intersect at a single point. But how do we determine if a system has no solution or an infinite number of solutions?

  • No Solution: If, during the substitution process, you arrive at a contradiction (e.g., 0 = 5), the system has no solution. This means the lines are parallel and never intersect.
  • Infinite Number of Solutions: If, after substitution and simplification, you end up with an identity (e.g., 0 = 0), the system has an infinite number of solutions. This means the two equations represent the same line, and every point on the line is a solution.

Why the Substitution Method Works

The effectiveness of the substitution method lies in its core principle: replacing an expression with its equivalent. In a system of equations, we're essentially looking for values that satisfy all equations simultaneously. When we solve one equation for a variable, we're defining that variable in terms of the others. By substituting this expression into another equation, we're not changing the solution set of the system. Instead, we're transforming the problem into a simpler, equivalent form.

The method's success hinges on the fact that if two expressions are equal, replacing one with the other doesn't alter the equation's validity. This transformation allows us to reduce the complexity of the system systematically. The substitution method is particularly powerful when one equation is already solved for a variable or when isolating a variable is relatively straightforward. However, the method's versatility extends to more complex systems as well.

Practice Makes Perfect

Solving systems of equations is a crucial skill in mathematics, with applications spanning various fields. To master the substitution method, consistent practice is key. Work through a variety of examples, varying in complexity and structure. Pay close attention to the algebraic manipulations involved, ensuring each step is logically sound. The more you practice, the more comfortable and confident you'll become in applying the method. Remember, the goal is not just to find the solution but to understand the underlying principles and the logic behind each step.

Conclusion

The substitution method is a reliable way to solve systems of equations. By following a step-by-step approach, you can systematically find the solutions and determine the nature of the solution set. In our example, the system has one solution, (-6, 7). Remember to practice and explore different systems to strengthen your understanding. For further learning and practice, you can explore resources like Khan Academy's Systems of Equations Section.