Solving Linear Systems: A Step-by-Step Guide

by ADMIN 45 views

Hey guys! Today, we're going to dive into the world of linear systems and tackle the question: How can I solve the linear system x - 7y = -3 and -5x - 6y = 15? Don't worry, it might look intimidating at first, but we'll break it down into easy-to-follow steps. There are several methods to solve linear systems, and we'll explore a couple of the most common ones: substitution and elimination. So, grab your pencils, and let's get started!

Understanding Linear Systems

Before we jump into solving, let's make sure we're all on the same page about what a linear system actually is. A linear system is simply a set of two or more linear equations that we're trying to solve simultaneously. Each equation represents a straight line when graphed, and the solution to the system is the point (or points) where these lines intersect. This intersection point represents the values of the variables (in our case, x and y) that satisfy all the equations in the system. There are three possible outcomes when solving a linear system:

  • One unique solution: The lines intersect at a single point.
  • No solution: The lines are parallel and never intersect.
  • Infinitely many solutions: The lines are the same, overlapping each other.

In our example, we have two equations:

  1. x - 7y = -3
  2. -5x - 6y = 15

Our goal is to find the values of x and y that make both of these equations true at the same time. Let's dive into our first method: substitution.

Method 1: Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable, leaving us with a single equation in one variable that we can easily solve. Here's how it works for our system:

Step 1: Solve one equation for one variable.

Looking at our equations, the first one (x - 7y = -3) seems easier to solve for x. Let's isolate x:

x - 7y = -3 x = 7y - 3

Great! We now have an expression for x in terms of y. This is a crucial step, so make sure you get this right.

Step 2: Substitute the expression into the other equation.

Now we'll substitute our expression for x (7y - 3) into the second equation (-5x - 6y = 15). This is where the magic happens! We're replacing x with an equivalent expression, so we don't change the solution to the system:

-5(7y - 3) - 6y = 15

Step 3: Solve the resulting equation.

We now have an equation with only one variable, y. Let's simplify and solve for y:

-35y + 15 - 6y = 15 -41y + 15 = 15 -41y = 0 y = 0

Awesome! We've found that y = 0. This is half of our solution. Now, let's find x.

Step 4: Substitute the value back into either original equation to solve for the other variable.

We can substitute y = 0 into either of our original equations or the expression we found for x in Step 1. Let's use the expression x = 7y - 3, as it's already set up to solve for x:

x = 7(0) - 3 x = -3

Fantastic! We've found that x = -3.

Step 5: Check your solution.

It's always a good idea to check your solution by plugging the values of x and y back into both original equations to make sure they hold true:

  • Equation 1: x - 7y = -3 --> -3 - 7(0) = -3 --> -3 = -3 (True)
  • Equation 2: -5x - 6y = 15 --> -5(-3) - 6(0) = 15 --> 15 = 15 (True)

Since our solution works in both equations, we're confident that it's correct!

The Solution

The solution to the linear system is x = -3 and y = 0, which can be written as the ordered pair (-3, 0). This is the point where the two lines represented by our equations intersect.

Method 2: Elimination

The elimination method (also sometimes called the addition method) involves manipulating the equations so that the coefficients of one of the variables are opposites. When we add the equations together, that variable is eliminated, leaving us with a single equation in one variable. Let's apply this method to our system:

Step 1: Multiply one or both equations by a constant so that the coefficients of one variable are opposites.

Looking at our equations:

  1. x - 7y = -3
  2. -5x - 6y = 15

We can eliminate x by multiplying the first equation by 5. This will give us a 5x term in the first equation, which is the opposite of the -5x term in the second equation:

5 * (x - 7y) = 5 * (-3) 5x - 35y = -15

Now our system looks like this:

  1. 5x - 35y = -15
  2. -5x - 6y = 15

Step 2: Add the equations together.

Now we add the two equations together. Notice how the x terms cancel out:

(5x - 35y) + (-5x - 6y) = -15 + 15 -41y = 0

Step 3: Solve the resulting equation.

We now have a single equation in y, which we can easily solve:

-41y = 0 y = 0

Just like with the substitution method, we found that y = 0!

Step 4: Substitute the value back into either original equation to solve for the other variable.

We substitute y = 0 back into either of the original equations. Let's use the first one:

x - 7(0) = -3 x = -3

Again, we find that x = -3.

Step 5: Check your solution.

We already checked this solution using the substitution method, and we know it works!

The Solution

The elimination method also gives us the solution x = -3 and y = 0, or the ordered pair (-3, 0). This confirms our answer from the substitution method.

Choosing a Method

So, which method should you use: substitution or elimination? The best method often depends on the specific system of equations. Here are some general guidelines:

  • Substitution: This method is often easier when one of the equations is already solved (or can be easily solved) for one variable. In our example, the first equation was easily solved for x, making substitution a good choice.
  • Elimination: This method is often easier when the coefficients of one variable are already opposites or can be easily made opposites by multiplying one or both equations by a constant. If you notice that adding or subtracting the equations will eliminate a variable, elimination might be the way to go.

Ultimately, both methods will lead you to the correct solution if applied correctly. Practice with different systems to get a feel for which method you prefer and which is most efficient for a given problem.

Graphical Interpretation

It's also helpful to visualize what's happening when we solve a linear system. Each equation represents a line on a graph. The solution to the system is the point where the lines intersect. In our case, the lines x - 7y = -3 and -5x - 6y = 15 intersect at the point (-3, 0). If the lines were parallel, there would be no solution. If the lines were the same, there would be infinitely many solutions.

Conclusion

Solving linear systems is a fundamental skill in algebra, and mastering it opens the door to more advanced mathematical concepts. We've explored two powerful methods: substitution and elimination. Remember to choose the method that seems most efficient for the given system, and always check your solution! By understanding the underlying principles and practicing regularly, you'll become a pro at solving linear systems in no time. Keep practicing, and don't hesitate to ask questions if you get stuck. You got this! This detailed guide should give you a strong understanding of how to solve systems of linear equations. Feel free to try it out on other examples and solidify your skills. Good luck, and happy solving!