Solving 2x + Y = 4 And X - Y = 1 A Step-by-Step Guide

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Hey guys! Today, we're diving into the fascinating world of solving systems of equations. Specifically, we're going to tackle the system:

  • 2x + y = 4
  • x - y = 1

We'll explore different methods to find the values of x and y that satisfy both equations simultaneously. Get ready to put on your math hats and let's get started!

Why Solve Systems of Equations?

Before we jump into the nitty-gritty, let's quickly discuss why solving systems of equations is important. You might be wondering, “When will I ever use this in real life?” Well, the truth is, systems of equations pop up in various fields, including:

  • Science: Calculating chemical reactions, determining forces in physics problems, and modeling population growth.
  • Engineering: Designing structures, analyzing circuits, and optimizing processes.
  • Economics: Predicting market trends, balancing supply and demand, and making financial forecasts.
  • Computer Science: Developing algorithms, creating simulations, and solving optimization problems.

So, mastering this skill is definitely worth your time and effort! It's a foundational concept that will help you in many areas.

Methods for Solving Systems of Equations

There are several methods we can use to solve systems of equations. We'll focus on three popular techniques:

  1. Substitution Method: This involves solving one equation for one variable and substituting that expression into the other equation.
  2. Elimination Method: This involves adding or subtracting the equations to eliminate one variable.
  3. Graphical Method: This involves plotting the equations on a graph and finding the point of intersection.

Let's explore each method in detail and see how they apply to our system of equations.

1. The Substitution Method: A Step-by-Step Approach

The substitution method is a powerful technique for solving systems of equations. The core idea is to isolate one variable in one equation and then substitute the expression for that variable into the other equation. This leaves you with a single equation with one variable, which you can easily solve. Let's break down the steps:

Step 1: Choose an Equation and Isolate a Variable

Look at the given equations:

  • 2x + y = 4
  • x - y = 1

We want to choose an equation where it's easy to isolate one of the variables. In this case, the second equation, x - y = 1, seems like a good candidate. We can easily isolate x by adding y to both sides:

  • x = y + 1

Now we have an expression for x in terms of y.

Step 2: Substitute the Expression into the Other Equation

We've isolated x in the second equation. Now, we'll substitute this expression (x = y + 1) into the first equation:

  • 2x + y = 4

Replace x with (y + 1):

  • 2(y + 1) + y = 4

Step 3: Solve the Resulting Equation

Now we have an equation with only one variable, y. Let's solve for y:

  • 2(y + 1) + y = 4
  • 2y + 2 + y = 4 (Distribute the 2)
  • 3y + 2 = 4 (Combine like terms)
  • 3y = 2 (Subtract 2 from both sides)
  • y = 2/3 (Divide both sides by 3)

Great! We've found the value of y: y = 2/3.

Step 4: Substitute the Value Back to Find the Other Variable

Now that we know y = 2/3, we can substitute this value back into either of the original equations or the expression we found in Step 1 (x = y + 1) to find x. Let's use the expression x = y + 1:

  • x = y + 1
  • x = (2/3) + 1
  • x = (2/3) + (3/3)
  • x = 5/3

So, we've found the value of x: x = 5/3.

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 the original equations:

  • 2x + y = 4

  • 2(5/3) + (2/3) = 4

  • (10/3) + (2/3) = 4

  • (12/3) = 4

  • 4 = 4 (This is true!)

  • x - y = 1

  • (5/3) - (2/3) = 1

  • (3/3) = 1

  • 1 = 1 (This is also true!)

Since our solution satisfies both equations, we can be confident that it's correct.

Solution

The solution to the system of equations is x = 5/3 and y = 2/3. We can write this as an ordered pair: (5/3, 2/3).

2. The Elimination Method: A Head-On Approach

The elimination method, also known as the addition or subtraction method, offers another powerful way to solve systems of equations. The beauty of this method lies in its direct approach: we manipulate the equations so that when we add or subtract them, one of the variables disappears, leaving us with a single equation in one variable. Let's break it down:

Step 1: Align the Equations and Identify a Variable to Eliminate

Write the equations one below the other, aligning the x terms, y terms, and constants:

  • 2x + y = 4
  • x - y = 1

Now, look for a variable that has coefficients that are either the same or opposites. In this case, the y terms have coefficients of +1 and -1, which are opposites. This is perfect for elimination!

Step 2: Multiply Equations (If Necessary) to Create Matching or Opposite Coefficients

In this example, the y coefficients are already opposites, so we don't need to multiply any equations. However, if the coefficients weren't opposites, we'd need to multiply one or both equations by a constant to make them opposites or identical. For instance, if we wanted to eliminate x, we could multiply the second equation by -2.

Step 3: Add or Subtract the Equations to Eliminate a Variable

Since the y coefficients are opposites, we'll add the two equations together:

2x + y = 4

  • x - y = 1

3x + 0 = 5

The y terms cancel out, leaving us with:

  • 3x = 5

Step 4: Solve for the Remaining Variable

Now we have a simple equation with one variable, x. Let's solve for x:

  • 3x = 5
  • x = 5/3

We've found the value of x: x = 5/3.

Step 5: Substitute the Value Back to Find the Other Variable

Now that we know x = 5/3, we can substitute this value back into either of the original equations to find y. Let's use the first equation, 2x + y = 4:

  • 2x + y = 4
  • 2(5/3) + y = 4
  • (10/3) + y = 4
  • y = 4 - (10/3)
  • y = (12/3) - (10/3)
  • y = 2/3

So, we've found the value of y: y = 2/3.

Step 6: Check Your Solution

As always, let's check our solution by plugging the values of x and y back into the original equations:

  • 2x + y = 4

  • 2(5/3) + (2/3) = 4

  • (10/3) + (2/3) = 4

  • (12/3) = 4

  • 4 = 4 (This is true!)

  • x - y = 1

  • (5/3) - (2/3) = 1

  • (3/3) = 1

  • 1 = 1 (This is also true!)

Our solution satisfies both equations, so we're good to go.

Solution

The solution to the system of equations is x = 5/3 and y = 2/3, or as an ordered pair: (5/3, 2/3).

3. The Graphical Method: Visualizing the Solution

The graphical method provides a visual way to solve systems of equations. Instead of manipulating equations algebraically, we graph each equation on the coordinate plane. The solution to the system is the point where the lines intersect. This method is particularly helpful for understanding the concept of solutions and for visualizing the relationship between the equations. Let's see how it works for our system:

Step 1: Rewrite the Equations in Slope-Intercept Form (y = mx + b)

To graph the equations easily, we'll rewrite them in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Let's start with the first equation:

  • 2x + y = 4
  • y = -2x + 4 (Subtract 2x from both sides)

Now, let's rewrite the second equation:

  • x - y = 1
  • -y = -x + 1 (Subtract x from both sides)
  • y = x - 1 (Multiply both sides by -1)

So, our equations in slope-intercept form are:

  • y = -2x + 4
  • y = x - 1

Step 2: Graph Each Equation

Now, we'll graph each equation on the coordinate plane. For the first equation, y = -2x + 4, the slope is -2 and the y-intercept is 4. For the second equation, y = x - 1, the slope is 1 and the y-intercept is -1. To graph each line, we can plot the y-intercept and then use the slope to find another point. Alternatively, we can create a table of values for x and y for each equation.

Step 3: Find the Point of Intersection

The solution to the system is the point where the two lines intersect. By looking at the graph, we can estimate the point of intersection. In this case, the lines intersect at approximately (1.67, 0.67), which is equivalent to (5/3, 2/3).

Step 4: Check Your Solution

To confirm our graphical solution, we can plug the coordinates of the point of intersection back into the original equations:

  • 2x + y = 4

  • 2(5/3) + (2/3) = 4

  • (10/3) + (2/3) = 4

  • (12/3) = 4

  • 4 = 4 (This is true!)

  • x - y = 1

  • (5/3) - (2/3) = 1

  • (3/3) = 1

  • 1 = 1 (This is also true!)

Since our solution satisfies both equations, we've successfully solved the system graphically.

Solution

The solution to the system of equations, as determined graphically, is approximately x = 5/3 and y = 2/3, or as an ordered pair: (5/3, 2/3). While the graphical method might not always give you the exact answer due to estimation, it provides a great visual understanding of the solution.

Conclusion: Mastering Systems of Equations

Alright, guys! We've explored three different methods for solving systems of equations: substitution, elimination, and graphical. We tackled the system 2x + y = 4 and x - y = 1 using each method and arrived at the same solution: x = 5/3 and y = 2/3. Remember, each method has its strengths and weaknesses, so it's a good idea to be comfortable with all three.

By understanding and practicing these techniques, you'll be well-equipped to tackle a wide range of problems involving systems of equations. Keep practicing, and you'll become a pro in no time!