Solving Systems Of Equations X + 2y = 4 And 2x + 4y = 8

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Solving systems of equations is a fundamental concept in algebra, and understanding how to approach these problems is crucial for various mathematical and real-world applications. In this comprehensive guide, we will delve into the specific system of equations:

x + 2y = 4
2x + 4y = 8

We will explore different methods to solve this system, analyze the nature of the solution, and discuss the implications of the results. Whether you're a student learning algebra or someone looking to refresh your knowledge, this guide will provide a clear and thorough understanding of how to solve this particular system of equations.

Understanding Systems of Equations

Before we dive into the specifics of solving the system x + 2y = 4 and 2x + 4y = 8, let's first establish a solid understanding of what a system of equations is and the different types of solutions that can arise. A system of equations is a set of two or more equations containing the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Graphically, each equation in the system represents a line (in the case of two variables), and the solution to the system corresponds to the point(s) where the lines intersect.

There are three possible outcomes when solving a system of two linear equations:

  1. Unique Solution: The lines intersect at a single point. This means there is one unique pair of values for x and y that satisfies both equations.
  2. No Solution: The lines are parallel and never intersect. This indicates that there is no pair of values for x and y that can simultaneously satisfy both equations. The system is said to be inconsistent.
  3. Infinitely Many Solutions: The lines are coincident, meaning they overlap completely. This implies that there are infinitely many pairs of values for x and y that satisfy both equations. The system is said to be dependent.

Understanding these possibilities is crucial because it helps us interpret the results we obtain when solving a system of equations. In the case of our system, x + 2y = 4 and 2x + 4y = 8, we will see that it falls into the third category – infinitely many solutions. This means that the two equations essentially represent the same line, just in different forms.

Methods for Solving Systems of Equations

There are several methods available for solving systems of equations, each with its own strengths and weaknesses. For the system x + 2y = 4 and 2x + 4y = 8, we will explore two common methods:

  • Substitution
  • Elimination

Method 1: Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be easily solved. Let's apply this method to our system:

  1. Solve one equation for one variable: Choose the first equation, x + 2y = 4, and solve for x:

    x = 4 - 2y

  2. Substitute: Substitute this expression for x into the second equation, 2x + 4y = 8:

    2(4 - 2y) + 4y = 8

  3. Simplify and solve for y: Distribute and simplify the equation:

    8 - 4y + 4y = 8
8 = 8

    Notice that the y terms cancel out, and we are left with the statement 8 = 8. This is a true statement, but it doesn't give us a specific value for y. This indicates that there are infinitely many solutions.

  4. Interpret the result: The equation 8 = 8 is always true, regardless of the value of y. This means that for any value of y, we can find a corresponding value of x that satisfies both equations. This confirms that the system has infinitely many solutions.

Method 2: Elimination

The elimination method involves manipulating the equations so that the coefficients of one of the variables are opposites. Then, by adding the equations together, that variable is eliminated, leaving a single equation with one variable. Let's apply this method to our system:

  1. Multiply equations to create opposite coefficients: Notice that the coefficient of x in the second equation is twice the coefficient of x in the first equation. To eliminate x, we can multiply the first equation by -2:

    -2(x + 2y) = -2(4)
-2x - 4y = -8

  2. Add the equations: Now, add the modified first equation to the second equation:

    (-2x - 4y) + (2x + 4y) = -8 + 8
0 = 0

  3. Interpret the result: Again, we obtain a true statement, 0 = 0, which doesn't give us specific values for x or y. This confirms that the system has infinitely many solutions.

Graphical Interpretation

To further understand the nature of the solution, let's visualize the system of equations graphically. Each equation represents a line in the coordinate plane. To graph the lines, we can find two points on each line or rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

  1. Rewrite the equations in slope-intercept form:

    • Equation 1: x + 2y = 4

      2y = -x + 4
y = -1/2x + 2

    • Equation 2: 2x + 4y = 8

      4y = -2x + 8
y = -1/2x + 2

  2. Analyze the equations: Notice that both equations have the same slope (-1/2) and the same y-intercept (2). This means that the two equations represent the same line. When graphed, they will overlap completely.

  3. Graphical conclusion: The fact that the two lines are coincident visually confirms that the system has infinitely many solutions. Any point on the line represents a solution to the system.

Implications of Infinitely Many Solutions

When a system of equations has infinitely many solutions, it implies that the equations are dependent. This means that one equation can be obtained from the other by multiplying it by a constant. In our case, the second equation 2x + 4y = 8 is simply the first equation x + 2y = 4 multiplied by 2.

This dependency has significant implications for the solution set. Instead of a unique solution (a single point) or no solution (parallel lines), the solution set is the entire line. This means that any pair of values (x, y) that satisfies one equation will also satisfy the other. To express the solution set, we can write it in terms of a parameter. For example, we can express x in terms of y (or vice versa) using one of the equations.

From the first equation, x + 2y = 4, we have x = 4 - 2y. We can let y be a parameter, say t, where t can be any real number. Then, x = 4 - 2t. The solution set can be written as:

{(x, y) | x = 4 - 2t, y = t, t ∈ ℝ}

This means that for any real number t, the pair (4 - 2t, t) is a solution to the system. For example:

  • If t = 0, then x = 4 and y = 0, giving the solution (4, 0).
  • If t = 1, then x = 2 and y = 1, giving the solution (2, 1).
  • If t = 2, then x = 0 and y = 2, giving the solution (0, 2).

And so on. There are infinitely many solutions, all lying on the same line.

Conclusion

In conclusion, the system of equations x + 2y = 4 and 2x + 4y = 8 has infinitely many solutions. We demonstrated this using two algebraic methods: substitution and elimination. Both methods led to the true statement 0 = 0 or 8 = 8, indicating dependency and infinite solutions. Furthermore, the graphical interpretation confirmed that the two equations represent the same line, providing a visual representation of the infinite solution set.

Understanding how to solve systems of equations and interpret the results is a crucial skill in algebra and beyond. Recognizing when a system has infinitely many solutions, as in this case, is just as important as finding a unique solution or determining that no solution exists. By mastering these concepts, you will be well-equipped to tackle more complex mathematical problems and real-world applications that involve systems of equations. This comprehensive guide has provided you with the tools and understanding necessary to confidently solve similar problems in the future. Remember to practice these methods and explore different systems of equations to further solidify your knowledge. The key to success in mathematics is consistent effort and a willingness to explore and understand the underlying concepts.