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How to solve a system of equations with two unknowns

How to solve a system of equations with two unknowns

The equation - is the identity, where among the famousmembers hiding one number, which is necessary to put in place the Latin letter, to the left and right side received the same numerical expression. To find it, you need to move in the same direction all the known members of the other - all the unknown terms of the equation.

And how to solve a system of two such equations? Individually - you can not, you should link the unknown quantities of the system with each other.

You can do this in three ways: by substitution, by the addition and the method of charting.

instructions

1

addition method.
two equations should be written strictly under each other:

-5u 2 = 61

-9h + 5y = -40.

Next, fold each term of equations, respectively, considering their signs:

2x + (- 9X) = - 7x, -5u + 5y = 0, 61 + (- 40) = 21. Typically, one of the sums of containing an unknown amount is zero.
Find the equation of the resulting members:

-7h + 0 = 21.
Find the unknown: -7h = 21, B = 21: (- 7) = - 3.
Substitute already found value in either of the original equations and get a second unknown, linear equation solving:

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

2

substitution method.
From one equation should express any of the required members:

x-5y = 61

-9h + 4y = -7.

x + 5y = 61, x = 61 + 5y.
To substitute the resulting equation into the second, instead of "X" (in this case):

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

linear equation to find the number "y":

-549 + 45u + 4y = -7, 45u + 4y = 549-7, 49u = 542, y = 542: 49, 11?.
In randomly selected (from the system) instead of the equation to insert already found "y" number of 11, and calculate the second unknown:

X = 61 * 11 + 5, x = 61 + 55 = 116 x.
This response system of equations: x = 116, y = 11.

3

Graphic method.
It consists in finding a practical locationthe point where the lines intersect, mathematically recorded in the system of equations. It is necessary to draw the graphs of both direct individually in the same coordinate system. General view of the line equation: - y = kx + b. To build the line, it suffices to find the coordinates of two points, and x is chosen randomly.
Consider a system: 2x - y = 4

-3h + y = 1.
Built right on the first equation forconvenience, it should be written: y = 2x-4. Come up with (easier) values ​​for X, substituting it into the equation, solving it, to find y. We have two points, which direct construction. (See fig.)
x 0 1

at -2 -4
Built right on the second equation: y = 1 + -3h.
Just to build the line. (See fig.)

0 x 2

from 1 to 5
Find the coordinates of the point of intersection of two straight lines constructed on the chart (if the lines do not intersect, then the system has no solution - it happens).

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