Solving Systems Of Linear Equations Step-by-Step Guide

Finding the solution to a system of linear equations is a fundamental concept in mathematics with applications spanning various fields. In this article, we will delve into a specific example, providing a comprehensive guide on how to solve it step-by-step. Our focus will be on a system of two linear equations with two variables, demonstrating the substitution method to arrive at the solution. Understanding these concepts is crucial for anyone studying algebra or related subjects.

The System of Equations

Let's consider the system of linear equations presented:

$\begin{aligned} 3y - 2x &= -9 \\ y &= -2x + 5 \end{aligned}$

Our goal is to find the values of x and y that satisfy both equations simultaneously. These values represent the point of intersection of the two lines represented by the equations.

Method 1: Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with a single variable, which can be easily solved. Here’s how it applies to our system:

Step 1: Identify the Solved Equation

Notice that the second equation, y = -2x + 5, is already solved for y. This makes it an ideal candidate for substitution. We can directly substitute this expression for y into the first equation.

Step 2: Substitute

Substitute the expression for y from the second equation into the first equation:

3(-2x + 5) - 2x = -9

This substitution replaces y in the first equation with its equivalent expression in terms of x. We now have a single equation with only one variable, x.

Step 3: Simplify and Solve for x

Now, we simplify the equation and solve for x:

• Distribute the 3: -6x + 15 - 2x = -9
• Combine like terms: -8x + 15 = -9
• Subtract 15 from both sides: -8x = -24
• Divide both sides by -8: x = 3

We have successfully found the value of x, which is 3. This is one part of the solution to our system of equations.

Step 4: Substitute the Value of x to Find y

Now that we know x = 3, we can substitute this value back into either of the original equations to find y. The second equation, y = -2x + 5, is simpler, so we'll use that:

y = -2(3) + 5 y = -6 + 5 y = -1

Therefore, the value of y is -1. We now have both x and y values.

Step 5: Write the Solution

The solution to the system of equations is the ordered pair (x, y), which represents the point where the two lines intersect. In this case, the solution is (3, -1).

Verification of the Solution

To ensure our solution is correct, we should verify it by substituting the values of x and y back into both original equations:

Equation 1: 3y - 2x = -9

Substitute x = 3 and y = -1:

3(-1) - 2(3) = -3 - 6 = -9

The equation holds true.

Equation 2: y = -2x + 5

Substitute x = 3 and y = -1:

-1 = -2(3) + 5 = -6 + 5 = -1

This equation also holds true. Since the solution (3, -1) satisfies both equations, we can confidently say that it is the correct solution to the system.

Alternative Methods for Solving Systems of Equations

While we used the substitution method in this example, there are other methods available for solving systems of linear equations. These include:

Elimination Method

The elimination method involves manipulating the equations so that when they are added or subtracted, one of the variables is eliminated. This results in a single equation with one variable, which can be solved. The value of the eliminated variable can then be found by substituting the result back into one of the original equations. For instance, consider these two equations:

$\begin{aligned} 2x + y &= 7 \\ x - y &= 2 \end{aligned}$

Here, adding the two equations directly eliminates y:

$\begin{aligned} (2x + y) + (x - y) &= 7 + 2 \\ 3x &= 9 \\ x &= 3 \end{aligned}$

Substituting x = 3 into the second equation gives:

3 - y = 2 y = 1

Thus, the solution is (x, y) = (3, 1).

Graphical Method

The graphical method involves plotting both equations on a coordinate plane. The solution to the system is the point where the two lines intersect. This method provides a visual representation of the solution. To solve the system graphically:

1. Rewrite each equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
2. Plot both lines on the same coordinate plane.
3. Identify the point of intersection. The coordinates of this point are the solution to the system.

For example, consider the system:

$\begin{aligned} y &= 2x + 1 \\ y &= -x + 4 \end{aligned}$

Plotting these lines reveals that they intersect at the point (1, 3). Therefore, the solution to the system is (x, y) = (1, 3).

Matrix Method

For more complex systems, especially those with three or more variables, matrix methods can be very efficient. This involves representing the system of equations as a matrix equation and using techniques such as Gaussian elimination or matrix inversion to solve for the variables. Consider the system:

$\begin{aligned} 2x + y - z &= 8 \\ x - y + z &= 1 \\ x + 2y + z &= 0 \end{aligned}$

This can be written in matrix form as:

$\begin{bmatrix} 2 & 1 & -1 \\ 1 & -1 & 1 \\ 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 8 \\ 1 \\ 0 \end{bmatrix}$

Solving this matrix equation using Gaussian elimination or matrix inversion will yield the values of x, y, and z.

Applications of Systems of Linear Equations

Systems of linear equations are not just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

Economics

In economics, systems of equations are used to model supply and demand curves. The intersection of these curves represents the market equilibrium, where the quantity supplied equals the quantity demanded.

Engineering

Engineers use systems of equations to analyze circuits, structural systems, and fluid flow. For example, in circuit analysis, Kirchhoff's laws lead to systems of linear equations that can be solved to find the currents and voltages in different parts of the circuit.

Computer Graphics

In computer graphics, systems of equations are used to perform transformations such as scaling, rotation, and translation of objects. These transformations are often represented using matrices, and solving systems of equations is necessary to apply these transformations accurately.

Chemistry

In chemistry, systems of equations are used to balance chemical equations and to solve stoichiometry problems. For example, balancing redox reactions often involves setting up and solving a system of equations.

Resource Allocation

Businesses and governments use linear programming, which involves solving systems of linear inequalities, to optimize resource allocation. This can include decisions about production levels, inventory management, and transportation logistics.

Conclusion

Solving systems of linear equations is a crucial skill in mathematics with broad applications across various disciplines. In this article, we have demonstrated how to solve a system of two linear equations using the substitution method. We also discussed other methods such as elimination, graphical, and matrix methods. Understanding these techniques allows you to tackle a wide range of problems in mathematics and beyond. Remember, the solution to a system of equations represents the point where the equations intersect, providing a valuable tool for problem-solving in many fields.

By mastering the methods for solving systems of linear equations, you gain a powerful tool for analyzing and solving problems in mathematics and various real-world applications. Whether you are a student learning the basics of algebra or a professional working in a technical field, understanding these concepts is essential for success.