Systems of two linear equations1 in two unknowns

Consider the equation

x — 2 y = 5 (1)

In this equation x=7 and y=1, but also x=5 and y=0. There are many such pairs of values which satisfy the equation (1). To find pairs other than those given, choose a value of one letter, say y arbitrary, and then from (1) find the сcorresponding value of x. For example, let y= 3. Then from (1)

x= 5 + 2y (2)

whence x=5+2*3, x— 5+6=11 and the pair of values x=11, y=3 satisfies the equation (1). The method for finding the pair of values satisfying both equations indicated above usually applies to pairs of equations of the form:

а₁x+ b₁у = c₁ (3)
а₂х + b₂у = c₂

where a_1, a_2, b_1, b₂, c_1, c₂ are known, and x and y are unknown quantities.

The equations (3) are termed linear because the unknown x and y enter to the first power only.

To solve a system of two linear equations in two unknowns, solve for one unknown in one equation and "substitute this result in the other equation, thus obtaining one equation in one unknown.

An alternative way² of solving a system of two linear equations, which is usually more convenient, is given by the following rule: multiply the two equations with numerical factors which are chosen so that³ the coefficient of one of the two unknowns have the same numerical values in both equations.

By adding or subtracting the two equations, a new equation with only one unknown quantity is obtained. Solve this equation. In order to find⁴ the second unknown quantity, substitute the value which has been found and solve for the
remaining unknown quantity. An alternative method for finding the second unknown is to repeat the above process of finding the equal coefficient for the other unknown.

Notes:

¹ equations in two unknowns — уравнения с двумя не
известными

² an alternative way — другой способ

³ so that — так что; таким образом, что

⁴ in order to find — для того, чтобы найти