How to solve systems of equations

It is not difficult to solve the system of equations by using the basic methods for solving systems of linear equations: the substitution method and the addition method.

Instruction manual

one

Let us consider methods for solving a system of equations using an example of a system of two linear equations having two unknown values. In general terms, such a system is written as follows (on the left, the equations are combined with a curly bracket):

ax + b = c

dx + ey = f, where

a, b, c, d, e, f are the coefficients (specific numbers), and x and y, as usual, are unknown. The numbers a, b, c, d are called the coefficients for unknowns, and c and f are called free terms. The solution to such a system of equations is found by two main methods.

Solving a system of equations by the substitution method

1. We take the first equation and express one of the unknowns (x) in terms of the coefficients and the other unknown (y):

x = (s-by) / a

2. Substitute the expression obtained for x into the second equation:

d (c-by) / a + ey = f

3. Solving the resulting equation, we find the expression for y:

y = (af-cd) / (ae-bd)

4. Substitute the resulting expression for y into the expression for x:

x = (ce-bf) / (ae-bd)

Example: you need to solve a system of equations:

3x-2y = 4

x + 3y = 5

Find the value of x from the first equation:

x = (2y + 4) / 3

Substitute the resulting expression into the second equation and get an equation with one variable (y):

(2y + 4) / 3 + 3y = 5, whence we get:

y = 1

Now we substitute the found value of y in the expression for the variable x:

x = (2 * 1 + 4) / 3 = 2

Answer: x = 2, y = 1.

2

The solution of the system of equations by the method of addition (subtraction).

This method reduces to multiplying both sides of the equations by numbers (parameters) such that, as a result, the coefficients of one of the variables coincide (possibly with the opposite sign).

In the general case, both sides of the first equation must be multiplied by (-d), and both sides of the second equation by a. As a result, we get:

-adx-bdу = -cd

adx + aey = af

Adding the resulting equations, we obtain:

-bdu + aeu = -cd + af, whence we get the expression for the variable y:

y = (af-cd) / (ae-bd), substituting the expression for y in any equation of the system, we obtain:

ax + b (af-cd) / (ae-bd) = c?

from this equation we find the second unknown:

x = (ce-bf) / (ae-bd)

Example. Solve the system of equations by adding or subtracting:

3x-2y = 4

x + 3y = 5

Multiply the first equation by (-1) and the second by 3:

-3x + 2y = -4

3x + 9y = 15

Adding (term by term) both equations, we obtain:

11y = 11

Where do we get:

y = 1

We substitute the obtained value for y into any of the equations, for example, into the second, we get:

3x + 9 = 15, whence

x = 2

Answer: x = 2, y = 1.