How to Solve a System of Equations: A Guide from Khan Academy

Solving a System of Equations: A Comprehensive Guide from Khan Academy

Solving asystem of equationscan be a daunting task for many students, but with the right approach, it can be a straightforward process. In this guide, we will explore the steps to solve a system of equations and provide examples to illustrate the process.

What is a System of Equations?

A system of equations is a set of two or more equations that have multiple unknown variables. The goal of solving a system of equations is to find the values of the unknown variables that satisfy all the equations in the system.

Types of Systems of Equations

There are three types of systems of equations:

1. Consistent: A system of equations is consistent if it has at least one solution that satisfies all the equations in the system.

2. Inconsistent: A system of equations is inconsistent if it has no solution that satisfies all the equations in the system.

3. Dependent: A system of equations is dependent if it has infinitely many solutions that satisfy all the equations in the system.

Solving a System of Equations

The process of solving a system of equations involves finding the values of the unknown variables that satisfy all the equations in the system. The steps to solve a system of equations are as follows:

1. Identify the type of system of equations.

2. Choose a method to solve the system of equations.

3. Apply the method to solve the system of equations.

4. Check the solution to ensure it satisfies all the equations in the system.

Methods to Solve a System of Equations

There are several methods to solve a system of equations, including:

1. Graphing: This method involves plotting each equation on a graph and finding the point where the lines intersect.

2. Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation.

3. Elimination: This method involves adding or subtracting the equations to eliminate one of the variables.

Examples

Let's look at some examples to illustrate the process of solving a system of equations.

Example 1:

Solve the following system of equations:

2x + 3y = 7

4x - y = 5

Step 1: Identify the type of system of equations.

This system of equations is consistent.

Step 2: Choose a method to solve the system of equations.

Let's use theeliminationmethod.

Step 3: Apply the method to solve the system of equations.

Multiplying the second equation by 3, we get:

12x - 3y = 15

Adding this equation to the first equation, we get:

14x = 22

Solving for x, we get:

x = 11/7

Substituting this value into the second equation, we get:

4(11/7) - y = 5

Solving for y, we get:

y = 13/7

Therefore, the solution to the system of equations is:

x = 11/7, y = 13/7

Step 4: Check the solution to ensure it satisfies all the equations in the system.

2(11/7) + 3(13/7) = 7

4(11/7) - 13/7 = 5

Both equations are satisfied, so the solution is correct.

Example 2:

Solve the following system of equations:

x - y = 3

2x + y = 5

Step 1: Identify the type of system of equations.

This system of equations is consistent.

Step 2: Choose a method to solve the system of equations.

Let's use thesubstitutionmethod.

Step 3: Apply the method to solve the system of equations.

Solving the first equation for y, we get:

y = x - 3

Substituting this expression into the second equation, we get:

2x + (x - 3) = 5

Solving for x, we get:

x = 4

Substituting this value into the first equation, we get:

4 - y = 3

Solving for y, we get:

y = 1

Therefore, the solution to the system of equations is:

x = 4, y = 1

Step 4: Check the solution to ensure it satisfies all the equations in the system.

4 - 1 = 3

2(4) + 1 = 5

Both equations are satisfied, so the solution is correct.

Conclusion

Solving a system of equations requires a systematic approach and the application of appropriate methods. By following the steps outlined in this guide and practicing with examples, students can become proficient in solving systems of equations.