Solving Linear Equations Finding Ordered Pair Solutions

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In the realm of mathematics, solving systems of linear equations is a fundamental skill. These systems, representing two or more equations with shared variables, pop up in various real-world scenarios, from balancing chemical equations to optimizing resource allocation. This article will delve into the methods to solve the system of linear equations, focusing on expressing the solution as an ordered pair, (x, y). We'll specifically address the system:

6x + 4y = 24
6x + 3y = 21

Let's embark on this journey to understand how to solve such systems effectively.

Understanding Systems of Linear Equations

Before diving into the solution, it's essential to grasp the concept of linear equations and their systems. A linear equation, at its core, is an algebraic equation where each term is either a constant or the product of a constant and a single variable. Graphically, a linear equation in two variables (x and y) represents a straight line on a coordinate plane. A system of linear equations, then, is a collection of two or more such equations that share the same variables. The solution to a system of linear equations is the set of values for the variables that satisfy all equations simultaneously. Geometrically, this solution corresponds to the point(s) where the lines represented by the equations intersect.

Solving a system of linear equations involves finding the values of the variables that make all the equations in the system true. In the case of two equations with two variables, the solution is typically an ordered pair (x, y), representing a single point on the coordinate plane. This point is the intersection of the two lines represented by the equations. There are several methods to solve systems of linear equations, each with its own strengths and weaknesses. The most common methods include graphing, substitution, and elimination. Graphing provides a visual representation of the equations and their intersection point, but it may not always yield precise solutions. Substitution involves solving one equation for one variable and substituting that expression into the other equation. This method can be particularly useful when one of the equations is already solved for a variable or can be easily manipulated to do so. Elimination, also known as the addition or subtraction method, involves manipulating the equations so that the coefficients of one variable are opposites, then adding the equations together to eliminate that variable. This method is often efficient when the coefficients of one variable are already opposites or can be easily made so.

Methods to Solve Linear Equations

There are multiple techniques available to solve systems of linear equations, each with its strengths and applicability depending on the specific system at hand. Understanding these methods empowers you to choose the most efficient approach for a given problem.

1. The Elimination Method

The elimination method, also known as the addition or subtraction method, is a powerful technique for solving systems of linear equations. The core idea behind this method is to manipulate the equations in the system so that the coefficients of one of the variables are opposites. Once this is achieved, adding the equations together will eliminate that variable, leaving you with a single equation in one variable. This equation can then be easily solved, and the solution can be substituted back into one of the original equations to find the value of the other variable. The elimination method is particularly effective when the coefficients of one variable are already opposites or can be easily made so by multiplying one or both equations by a constant.

To apply the elimination method effectively, follow these steps:

  1. Align the equations: Ensure that the like terms (terms with the same variable) are aligned vertically.
  2. Multiply equations (if necessary): If the coefficients of the variable you want to eliminate are not opposites, multiply one or both equations by a constant so that they become opposites. For example, if you want to eliminate 'x' and the coefficients of 'x' are 2 and 3, you could multiply the first equation by 3 and the second equation by -2 to get coefficients of 6 and -6.
  3. Add the equations: Add the equations together. This will eliminate one variable, leaving you with a single equation in one variable.
  4. Solve for the remaining variable: Solve the resulting equation for the remaining variable.
  5. Substitute back: Substitute the value you found in step 4 into one of the original equations and solve for the other variable.
  6. Check your solution: Substitute the values of both variables into both original equations to ensure that the solution satisfies both equations.

2. The Substitution Method

The substitution method provides another valuable approach to solving systems of linear equations. This technique involves solving one equation for one variable and then substituting that expression into the other equation. This substitution eliminates one variable, resulting in a single equation with one unknown. Solving this equation yields the value of one variable, which can then be substituted back into either of the original equations to find the value of the other variable. The substitution method shines when one of the equations is already solved for a variable or can be easily manipulated to do so.

The steps to implement the substitution method are as follows:

  1. Solve for one variable: Choose one of the equations and solve it for one of the variables. It's often advantageous to choose an equation where a variable has a coefficient of 1, as this simplifies the process.
  2. Substitute: Substitute the expression you obtained in step 1 into the other equation. This will result in an equation with only one variable.
  3. Solve for the remaining variable: Solve the equation from step 2 for the remaining variable.
  4. Substitute back: Substitute the value you found in step 3 back into either of the original equations (or the expression you found in step 1) to solve for the other variable.
  5. Check your solution: As with the elimination method, it's crucial to check your solution by substituting the values of both variables into both original equations to ensure they are satisfied.

3. The Graphing Method

The graphing method offers a visual approach to solving systems of linear equations. This technique involves graphing each equation in the system on the coordinate plane. The solution to the system corresponds to the point(s) where the lines intersect. If the lines intersect at a single point, the coordinates of that point represent the unique solution to the system. If the lines are parallel, they do not intersect, indicating that the system has no solution. If the lines coincide (are the same line), there are infinitely many solutions, as every point on the line satisfies both equations.

To solve a system of linear equations using the graphing method, follow these steps:

  1. Rewrite equations (optional): If necessary, rewrite the equations in slope-intercept form (y = mx + b), where 'm' is the slope and 'b' is the y-intercept. This form makes it easier to graph the lines.
  2. Graph the lines: Graph each equation on the same coordinate plane. You can do this by plotting two points on each line (e.g., the x- and y-intercepts) and drawing a line through them, or by using the slope and y-intercept to plot the line.
  3. Identify the intersection: Look for the point(s) where the lines intersect. The coordinates of the intersection point(s) represent the solution(s) to the system.
  4. Check your solution: Substitute the coordinates of the intersection point(s) into both original equations to verify that they satisfy both equations.

The graphing method provides a valuable visual understanding of the system of equations and its solution. However, it may not always yield precise solutions, especially if the intersection point has non-integer coordinates. In such cases, algebraic methods like substitution or elimination may be more accurate.

Solving the Given System: 6x + 4y = 24 and 6x + 3y = 21

Let's tackle the system of linear equations provided:

6x + 4y = 24
6x + 3y = 21

We'll demonstrate the elimination method here, as it proves to be particularly efficient for this system.

Applying the Elimination Method

  1. Align the equations: The equations are already aligned, with the 'x' terms, 'y' terms, and constants in their respective columns.
  2. Multiply equations (if necessary): Notice that the coefficients of 'x' are the same (both are 6). To eliminate 'x', we can multiply one of the equations by -1. Let's multiply the second equation by -1:
    -1 * (6x + 3y) = -1 * 21
-6x - 3y = -21
  3. Add the equations: Now, add the first equation (6x + 4y = 24) and the modified second equation (-6x - 3y = -21):
    (6x + 4y) + (-6x - 3y) = 24 + (-21)
6x - 6x + 4y - 3y = 3
y = 3
  4. Solve for the remaining variable: We have found that y = 3.
  5. Substitute back: Substitute y = 3 into either of the original equations to solve for 'x'. Let's use the first equation:
    6x + 4(3) = 24
6x + 12 = 24
6x = 12
x = 2
  6. Check your solution: Substitute x = 2 and y = 3 into both original equations:
    • Equation 1: 6(2) + 4(3) = 12 + 12 = 24 (Correct)
    • Equation 2: 6(2) + 3(3) = 12 + 9 = 21 (Correct)

Solution

The solution to the system of equations is x = 2 and y = 3. Therefore, the ordered pair solution is (2, 3).

Conclusion

Solving systems of linear equations is a crucial skill in mathematics. We've explored the elimination method and applied it to find the solution to the system:

6x + 4y = 24
6x + 3y = 21

We determined that the solution, expressed as an ordered pair, is (2, 3). Mastering these techniques empowers you to tackle a wide range of mathematical problems and real-world applications involving linear relationships.