# How to solve a system of equations with two unknowns

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An equation is an identity where among the knownMembers hides one number, which must be put instead of the Latin letter, so that the left and right sides get the same numerical expression. To find it, you need to transfer all known terms to one side, to the other - all the unknown terms of the equation.

And how to solve a system of two such equations? Separately - it is impossible, it is necessary to connect the required quantities from the system with each other.

You can do this in three ways: by substitution, by addition, and by plotting.

Instructions

- 1

The method of addition.

It is necessary to write down two equations strictly one under the other:

2 -5y = 61

-9x + 5y = -40.

Next, add up each term of the equations, respectively, taking into account their signs:

2x + (-9x) = -7x, -5y + 5y = 0, 61 + (-40) = 21. As a rule, one of the sums containing an unknown quantity will be zero.

Write an equation from the received members:

-7x + 0 = 21.

Find the unknown: -7x = 21, h = 21: (-7) = - 3.

Substitute the value found in any of the original equations and obtain the second unknown by solving the linear equation:

2x-5y = 61, 2 (-3) -5y = 61, -6-5y = 61, -5y = 61 + 6, -5y = 67, y = -13.4.

The answer is a system of equations: x = -3, y = -13.4.

- 2

The method of substitution.

From one equation it is necessary to express any of the required terms:

X-5y = 61

-9x + 4y = -7.

X = 61 + 5y, x = 61 + 5y.

Substitute the resulting equation into the second instead of the number "X" (in this case):

-9 (61 + 5y) + 4y = -7.

Further solving

Linear equation, find the number "game":

-549 + 45y + 4y = -7, 45y + 4y = 549-7, 49y = 542, y = 542: 49, y11.

In an arbitrarily chosen (from the system) equation, insert in place of the already found "game" the number 11 and calculate the second unknown:

X = 61 + 5 * 11, x = 61 + 55, x = 116.

The answer of this system of equations is: x = 116, y = 11.

- 3

Graphical method.

Is in the practical location of the coordinatesA point at which lines intersected mathematically recorded in the system of equations. It is necessary to draw the graphs of both straight lines separately in one coordinate system. The general form of the equation of the straight line: - y = kx + b. To construct a straight line, it is sufficient to find the coordinates of two points, where, x is chosen arbitrarily.

Let there be given a system: 2x = y = 4

Y = -3x + 1.

A straight line is constructed according to the first equation, forIts convenience must be written: y = 2x-4. To think up (lighter) the values for X, substituting it into the equation, solving it, to find a game. Two points are obtained on which the line is constructed. (See fig.)

X 0 1

Y -4 -2

A straight line is constructed according to the second equation: y = -3x + 1.

Just build a straight line. (See fig.)

Х 0 2

Y 1 -5

Find the coordinates of the intersection point of the two straight lines on the graph (if the lines do not intersect, then the system of equations does not have a solution - it happens).