Solving systems of algebraic equations with two unknowns is an essential skill in mathematics. Students learn it in algebra class, and it is used in various fields like physics, engineering, and even economics. It helps find the solutions to problems that involve two variables or quantities.

To solve systems of algebraic equations with two unknowns, we must follow specific steps. Here’s how to do it:

Step 1: Determine the type of system.

There are three types of systems of equations – consistent, inconsistent, and dependent. Inconsistent systems have no common solutions. Dependent systems have an infinite number of solutions. Consistent systems have a unique solution. To determine the type of system, we must analyze the equations and their coefficients.

Step 2: Choose a method to solve the system.

There are various methods to solve systems of algebraic equations with two unknowns. The four most common methods are: substitution, elimination, graphing, and matrices. Each method has its advantages and disadvantages, so we choose a method according to the type of system and the available resources.

Step 3: Solve the system using the chosen method.

A. Substitution Method

In this method, we substitute one equation’s variable in the other equation and solve for the other variable. Then, we substitute the solution back into one equation to find the other variable’s value.

For example, let’s solve the following system using substitution method:

x + y = 5
2x – y = 4

We can write the first equation as y = 5 – x. Substituting this value in the second equation, we get:

2x – (5 – x) = 4
3x = 9
x = 3

Substituting x = 3 in the first equation, we get:

3 + y = 5
y = 2

So the solution to the system is (x, y) = (3, 2).

B. Elimination Method

In this method, we eliminate one variable by adding or subtracting the two equations to create a new equation with only one variable. Then, we solve for the variable and substitute the solution back into one equation to find the other variable’s value.

For example, let’s solve the following system using elimination method:

2x – 3y = 7
4x + y = 1

Multiplying the first equation by 4 and the second equation by 3, we get:

8x – 12y = 28
12x + 3y = 3

Adding these two equations, we get:

20x = 31
x = 31/20

Substituting x = 31/20 in the first equation, we get:

2(31/20) – 3y = 7
y = -19/20

So the solution to the system is (x, y) = (31/20, -19/20).

C. Graphing Method

In this method, we graph the two equations on the same coordinate plane and find their intersection points, which are the solutions to the system.

For example, let’s solve the following system using graphing method:

y = x + 1
y = -2x + 5

Graphing these two equations on the same coordinate plane, we get:

The intersection point is (2, 3), which is the solution to the system.

D. Matrices Method

In this method, we use matrices to represent the coefficients of the variables and solve for their values using matrix algebra.

For example, let’s solve the following system using matrices method:

2x + y = 5
3x – 2y = -2

Writing the coefficients in a matrix form, we get:

Using matrix algebra, we can solve for the variables’ values as follows:

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So the solutions to the system are x = 5/2 and y = 11/2.

In conclusion, solving systems of algebraic equations with two unknowns is a crucial skill in mathematics. By following the necessary steps and choosing the appropriate method, we can easily find the solutions to various problems involving two variables.

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