Solving System Of Linear Equation By Substitution Method

Solving systems of linear equations is a fundamental skill in mathematics and has wide applications in various fields such as physics, engineering, economics, and computer science. One of the most straightforward methods for solving these systems is the substitution method.

What is the Substitution Method?

The substitution method is an algebraic technique used to solve a system of equations by expressing one variable in terms of another, and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, which can then be easily solved. The solution is then substituted back into one of the original equations to find the value of the other variable.

When to Use the Substitution Method

The substitution method is particularly useful when one of the equations in the system can be easily solved for one variable in terms of the other. This usually occurs when one of the variables has a coefficient of 1 or -1. However, the substitution method can be applied to any system of linear equations, regardless of the coefficients.

Steps to Solve System of Linear Equation by Substitution Method

Step 1: Solve One Equation for One Variable

Begin by selecting one of the equations in the system and solve it for one variable in terms of the other. Choose the equation and variable that make this step as simple as possible. For example, if one of the equations is x + y = 5, it is easy to solve for x as x = 5 - y.

Step 2: Substitute the Expression into the Other Equation

Take the expression obtained in step 1 and substitute it into the other equation in the system. This will result in a new equation that contains only one variable.

Step 3: Solve the New Equation

Solve the equation obtained in step 2 for the remaining variable. This will give you the numerical value of one of the variables in the system.

Step 4: Substitute Back to Find the Other Variable

Substitute the value obtained in step 3 back into one of the original equations (or the expression obtained in step 1) to find the value of the other variable.

Step 5: Check Your Solution

To ensure accuracy, substitute both values obtained into both original equations to verify that they satisfy both equations.

Example 1: A Simple System of Equations

Consider the system of equations:

• x + y = 5
• 2x - y = 1

Step 1: Solve One Equation for One Variable

From the first equation, we can easily solve for x in terms of y:

• x = 5 - y

Step 2: Substitute the Expression into the Other Equation

Substitute this expression for x into the second equation:

• 2(5 - y) - y = 1

Step 3: Solve the New Equation

Simplify and solve for y:

• 10 - 2y - y = 1
• 10 - 3y = 1
• -3y = -9
• y = 3

Step 4: Substitute Back to Find the Other Variable

Substitute y = 3 back into the expression for x:

• x = 5 - 3
• x = 2

Step 5: Check Your Solution

Check the solution in both original equations:

• Equation 1: 2 + 3 = 5 (True)
• Equation 2: 2(2) - 3 = 1 (True)

Therefore, the solution to the system of equations is x = 2 and y = 3.

Example 2: A More Complex System of Equations

Consider the system of equations:

• 3x + 2y = 11
• x - y = 3

Step 1: Solve One Equation for One Variable

From the second equation, we can easily solve for x in terms of y:

• x = y + 3

Step 2: Substitute the Expression into the Other Equation

Substitute this expression for x into the first equation:

• 3(y + 3) + 2y = 11

Step 3: Solve the New Equation

Simplify and solve for y:

• 3y + 9 + 2y = 11
• 5y + 9 = 11
• 5y = 2
• y = 2/5

Step 4: Substitute Back to Find the Other Variable

Substitute y = 2/5 back into the expression for x:

• x = (2/5) + 3
• x = (2/5) + (15/5)
• x = 17/5

Step 5: Check Your Solution

Check the solution in both original equations:

• Equation 1: 3(17/5) + 2(2/5) = 51/5 + 4/5 = 55/5 = 11 (True)
• Equation 2: (17/5) - (2/5) = 15/5 = 3 (True)

Therefore, the solution to the system of equations is x = 17/5 and y = 2/5.

Example 3: Dealing with Fractions

Consider the system of equations:

• (1/2)x + y = 4
• x - (1/3)y = 2

Step 1: Solve One Equation for One Variable

It might be easier to solve the second equation for x:

• x = (1/3)y + 2

Step 2: Substitute the Expression into the Other Equation

Substitute this expression for x into the first equation:

• (1/2)((1/3)y + 2) + y = 4

Step 3: Solve the New Equation

Simplify and solve for y:

• (1/6)y + 1 + y = 4
• (7/6)y = 3
• y = (3 * 6) / 7
• y = 18/7

Step 4: Substitute Back to Find the Other Variable

Substitute y = 18/7 back into the expression for x:

• x = (1/3)(18/7) + 2
• x = (6/7) + (14/7)
• x = 20/7

Step 5: Check Your Solution

Check the solution in both original equations:

• Equation 1: (1/2)(20/7) + (18/7) = (10/7) + (18/7) = 28/7 = 4 (True)
• Equation 2: (20/7) - (1/3)(18/7) = (20/7) - (6/7) = 14/7 = 2 (True)

Therefore, the solution to the system of equations is x = 20/7 and y = 18/7.

Special Cases

No Solution

Sometimes, when you substitute and simplify, you might end up with a contradiction, such as 0 = 1. This indicates that the system of equations has no solution, meaning the lines represented by the equations are parallel and never intersect.

Example:

• x + y = 3
• x + y = 5

If we solve the first equation for x, we get x = 3 - y. Substituting into the second equation:

• (3 - y) + y = 5
• 3 = 5 (Contradiction)

This system has no solution.

Infinite Solutions

If, after substituting and simplifying, you end up with an identity, such as 0 = 0, this indicates that the system of equations has infinitely many solutions. This means the two equations represent the same line.

Example:

• x + y = 2
• 2x + 2y = 4

If we solve the first equation for x, we get x = 2 - y. Substituting into the second equation:

• 2(2 - y) + 2y = 4
• 4 - 2y + 2y = 4
• 4 = 4 (Identity)

This system has infinitely many solutions.

Tips and Tricks

Choose the Easiest Variable to Solve For

Look for equations where one variable has a coefficient of 1 or -1. Solving for this variable will minimize the chances of dealing with fractions.

Be Careful with Signs

When substituting, pay close attention to the signs of the terms. A small mistake with a sign can lead to an incorrect solution.

Simplify Before Substituting

If possible, simplify the equations before you start the substitution process. This can make the calculations easier.

Check Your Work

Always check your solution by substituting the values back into the original equations. This will help you catch any mistakes.

Advantages of the Substitution Method

• Simplicity: The substitution method is relatively straightforward and easy to understand, making it a good choice for simple systems of equations.
• Versatility: It can be applied to any system of linear equations, regardless of the coefficients.
• Intuitive: The process of expressing one variable in terms of another is intuitive and easy to visualize.

Disadvantages of the Substitution Method

• Complexity with Fractions: The substitution method can become cumbersome when dealing with systems of equations that involve fractions or complex expressions.
• Not Ideal for Large Systems: For systems with more than two variables, the substitution method can become very complicated and time-consuming.

Applications of Solving Linear Equations

Solving systems of linear equations is not just an abstract mathematical exercise; it has numerous real-world applications. Here are a few examples:

Physics

In physics, systems of linear equations are used to solve problems involving motion, forces, and electrical circuits. For example, Kirchhoff's laws for electrical circuits can be expressed as a system of linear equations.

Engineering

Engineers use systems of linear equations to analyze structures, design control systems, and model fluid flow. For example, the stress and strain in a bridge can be calculated using a system of linear equations.

Economics

Economists use systems of linear equations to model supply and demand, analyze market equilibrium, and forecast economic trends. For example, the equilibrium price and quantity in a market can be determined by solving a system of linear equations.

Computer Science

In computer science, systems of linear equations are used in computer graphics, data analysis, and machine learning. For example, linear regression, a common technique in machine learning, involves solving a system of linear equations.

Substitution Method vs. Elimination Method

The substitution method is one way to solve systems of linear equations. Another popular method is the elimination method, also known as the addition method. In the elimination method, you manipulate the equations so that when you add or subtract them, one of the variables is eliminated. Here's a brief comparison:

Substitution Method

• Best Used When: One equation is easily solved for one variable.
• Process: Solve for one variable and substitute into the other equation.
• Advantages: Simple and intuitive.
• Disadvantages: Can be cumbersome with complex equations or large systems.

Elimination Method

• Best Used When: No equation is easily solved for one variable, or when coefficients are easily made opposites.
• Process: Manipulate equations to eliminate one variable by adding or subtracting equations.
• Advantages: Can be more efficient for some systems, especially larger ones.
• Disadvantages: May require more steps to set up.

Example Comparing the Methods

Consider the system:

• 2x + y = 7
• x - 2y = -4

Using Substitution:

1. Solve the second equation for x: x = 2y - 4
2. Substitute into the first equation: 2(2y - 4) + y = 7
3. Simplify and solve for y: 4y - 8 + y = 7 => 5y = 15 => y = 3
4. Substitute back to find x: x = 2(3) - 4 => x = 2

Using Elimination:

1. Multiply the second equation by -2: -2x + 4y = 8
2. Add the modified second equation to the first equation:
   • (2x + y) + (-2x + 4y) = 7 + 8
   • 5y = 15 => y = 3
3. Substitute back into one of the original equations: 2x + 3 = 7 => 2x = 4 => x = 2

In this case, both methods are equally efficient. However, for other systems, one method might be more straightforward than the other.

Conclusion

The substitution method is a powerful and versatile tool for solving systems of linear equations. By following a systematic approach and paying attention to detail, you can use this method to solve a wide variety of problems in mathematics, science, and engineering. While it may not always be the most efficient method for all systems of equations, its simplicity and intuitive nature make it a valuable technique to have in your mathematical toolkit. Remember to always check your solutions to ensure accuracy and to be aware of special cases such as systems with no solution or infinitely many solutions.