Solving 2x + 3y = 7 And X - Y = 1 A Step-by-Step Guide

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Hey everyone! Today, we're diving into the world of linear equations and tackling a classic problem: solving a system of two equations. Don't worry if the term sounds intimidating; we'll break it down into easy-to-follow steps. We're going to solve the following system:

2x + 3y = 7
x - y = 1

This is a common type of problem in algebra, and mastering it will give you a solid foundation for more advanced math concepts. So, let's get started!

Understanding Systems of Linear Equations

Before we jump into solving, let's quickly understand what we're dealing with. A system of linear equations is simply a set of two or more linear equations that involve the same variables. In our case, we have two equations, and both involve the variables x and y. The goal is to find the values of x and y that satisfy both equations simultaneously. Think of it like finding the point where two lines intersect on a graph. That point represents the solution to the system.

Each equation in our system represents a straight line when graphed. The solution to the system is the point where these lines intersect. If the lines are parallel, there's no solution (they never intersect). If the lines are the same, there are infinitely many solutions (they overlap completely).

Solving systems of equations is a fundamental skill in various fields, including mathematics, physics, engineering, economics, and computer science. They're used to model real-world situations, optimize processes, and make predictions. For example, you might use a system of equations to determine the break-even point for a business, calculate the trajectory of a projectile, or analyze the flow of traffic in a network. So, learning how to solve them is definitely worth your time!

There are several methods for solving systems of linear equations, and we'll focus on two popular ones: substitution and elimination. Let's explore the substitution method first.

Method 1: Solving by Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively eliminates one variable, leaving you with a single equation in one variable, which you can easily solve. Once you've found the value of one variable, you can substitute it back into either of the original equations to find the value of the other variable. Sound confusing? Don't worry; we'll walk through it step by step.

Step 1: Solve one equation for one variable.

Look at our system again:

2x + 3y = 7
x - y = 1

The second equation, x - y = 1, looks simpler to manipulate. Let's solve it for x. To do this, we add y to both sides of the equation:

x - y + y = 1 + y
x = 1 + y

Great! We've now expressed x in terms of y. This is the crucial first step in the substitution method. We've isolated x and have an expression that we can substitute into the other equation.

Step 2: Substitute the expression into the other equation.

Now, we take the expression we found for x, which is x = 1 + y, and substitute it into the other equation (the one we haven't used yet), which is 2x + 3y = 7. This means we replace the x in the first equation with the expression (1 + y):

2(1 + y) + 3y = 7

Notice what we've done: we've replaced x with an equivalent expression in terms of y. This is the heart of the substitution method. We've transformed the equation into one that only involves y, which we can then solve.

Step 3: Solve the resulting equation for the remaining variable.

Now we have the equation 2(1 + y) + 3y = 7. Let's simplify and solve for y. First, we distribute the 2:

2 + 2y + 3y = 7

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

2 + 5y = 7

Now, we subtract 2 from both sides:

5y = 5

Finally, we divide both sides by 5:

y = 1

Fantastic! We've found the value of y: y = 1. This is one half of our solution. We now know the y-coordinate of the point where the two lines intersect.

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

We now know that y = 1. To find the value of x, we can substitute this value back into either of the original equations. Let's use the simpler equation, x - y = 1:

x - 1 = 1

Adding 1 to both sides, we get:

x = 2

Excellent! We've found the value of x: x = 2. We now have both parts of our solution.

Step 5: Check your solution.

It's always a good idea to check your solution to make sure it works in both original equations. This helps prevent errors and ensures you have the correct answer.

Let's check our solution x = 2 and y = 1 in the first equation, 2x + 3y = 7:

2(2) + 3(1) = 4 + 3 = 7

It works! Now let's check in the second equation, x - y = 1:

2 - 1 = 1

It works there too! Our solution satisfies both equations, so we know we've found the correct answer.

Solution:

Therefore, the solution to the system of equations is x = 2 and y = 1. We can write this as an ordered pair: (2, 1). This represents the point where the two lines represented by the equations intersect on a graph.

Method 2: Solving by Elimination

The elimination method (also called the addition method) is another powerful technique for solving systems of linear equations. It involves manipulating the equations so that when you add them together, one of the variables is eliminated. This leaves you with a single equation in one variable, which you can solve. Then, you can substitute the value you found back into one of the original equations to find the value of the other variable. This method is particularly useful when the coefficients of one of the variables are opposites or can easily be made opposites.

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

Let's look at our system again:

2x + 3y = 7
x - y = 1

Notice that the coefficients of y are 3 and -1. We can easily make these opposites by multiplying the second equation by 3. This will give us a -3y term, which is the opposite of the 3y term in the first equation.

So, we multiply the entire second equation, x - y = 1, by 3:

3(x - y) = 3(1)
3x - 3y = 3

Now our system looks like this:

2x + 3y = 7
3x - 3y = 3

See how the y terms are now opposites? This is the key to the elimination method. We've set up the equations so that when we add them together, the y variable will disappear.

Step 2: Add the equations together.

Now we add the two equations together, term by term:

(2x + 3y) + (3x - 3y) = 7 + 3

Combining like terms, we get:

5x = 10

Notice that the y terms have canceled out, as planned! This is the beauty of the elimination method. We've successfully eliminated one variable and are left with a simple equation in one variable.

Step 3: Solve the resulting equation for the remaining variable.

We now have the equation 5x = 10. To solve for x, we divide both sides by 5:

x = 2

Great! We've found the value of x: x = 2. This is one part of our solution. We now need to find the value of y.

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

We know that x = 2. To find the value of y, we can substitute this value back into either of the original equations. Let's use the second equation, x - y = 1:

2 - y = 1

Subtracting 2 from both sides, we get:

-y = -1

Multiplying both sides by -1, we get:

y = 1

Excellent! We've found the value of y: y = 1. We now have both parts of our solution.

Step 5: Check your solution.

As before, let's check our solution x = 2 and y = 1 in both original equations.

In the first equation, 2x + 3y = 7:

2(2) + 3(1) = 4 + 3 = 7

It works!

In the second equation, x - y = 1:

2 - 1 = 1

It works there too! Our solution satisfies both equations, so we know we've found the correct answer.

Solution:

Therefore, the solution to the system of equations is x = 2 and y = 1, or (2, 1) as an ordered pair.

Comparing Substitution and Elimination

So, we've solved the same system of equations using two different methods: substitution and elimination. Which method is better? Well, it depends on the specific system you're dealing with.

  • Substitution is often a good choice when one of the equations is already solved for one variable or can be easily solved for one variable. In our example, the equation x - y = 1 was easily solved for x, making substitution a straightforward approach.
  • Elimination is often a good choice when the coefficients of one of the variables are opposites or can easily be made opposites by multiplying one or both equations by a constant. In our example, the y coefficients were easily made opposites, making elimination a convenient method.

Ultimately, the best method is the one you feel most comfortable with and that seems most efficient for the given problem. Practice with both methods, and you'll develop a sense of which one is best suited for different situations.

Key Takeaways

  • A system of linear equations consists of two or more linear equations with the same variables.
  • The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously.
  • The substitution method involves solving one equation for one variable and substituting that expression into the other equation.
  • The elimination method involves manipulating the equations so that when you add them together, one of the variables is eliminated.
  • Always check your solution by substituting the values back into the original equations.

Practice Makes Perfect

The best way to master solving systems of linear equations is to practice! Try solving different systems using both substitution and elimination. You'll encounter different scenarios and learn how to adapt your approach. There are plenty of resources online and in textbooks that offer practice problems. The more you practice, the more confident you'll become in your ability to solve these problems.

Solving systems of equations is a fundamental skill in algebra and has applications in many different fields. By understanding the concepts and practicing the techniques, you'll be well-equipped to tackle these problems and apply them to real-world situations. So, keep practicing, and you'll become a system-solving pro in no time!

I hope this guide has helped you understand how to solve systems of linear equations. Keep practicing, and you'll become a pro in no time! Good luck, guys!