Instruction

1

Convert one of the equations to the form in which y is expressed through x or Vice versa x through y. Substitute the resulting expression is (or y instead of x) into the second equation. You will get an equation with one variable.

2

To solve some systems of equations required to Express both variables x and y through one or two new variables. To do this, enter one

**variable**m for only one equation and two variables m and n for both equations.3

Example I. Express one

**variable**through the other in the system of equations:│x–2y=1,│x2+xy–y2=11.Convert the first equation of the system: transfer a single term (–2y) to the right side of the equality by changing the sign. Hence we get: x=1+2y.4

Substitute in the equation x2+xy–y2=11 is x 1+2y. The system of equations becomes:│(1+2y)2+(1+2y)y–y2=11,│x=1+2y.The obtained system is equivalent to the original. You have expressed

**the variable**x in the given system of equations using y.5

Example II. Express one

**variable**through the other in the system of equations:│x2–y2=5,│xy=6. Transform the second equation of the system: both parts of the equation xy=6 divide by x≠0. Hence: y=6/x.6

Substitute the resulting expression into the equation x2–y2=5. You will receive:│x2–(6/x)2=5,│y=6/x. The last system is equivalent to the original. You have expressed

**the variable**y in the system of equations in x.7

Example III. Express the variables y and z, using new variables m and n:│2/(y+z)+9/(2y+z)=2;│4/(y+z)=12/(2y+z) -1.Let 1/(y+z)=m, and 1/(2y+z)=n. Then the system of equations will look as follows:│2/m+9/n=2,│4/m=12/n–1.You have expressed the variables y and z into the original system equations in the new variables m and n.

Note

A welcome introduction of a new variable used in the solution of some quadratic equations. For example, in the equation (x2+1)/x+x/(x2+1)=-2,5 Express the variable x in a new variable y. Let y=(x2+1)/x, the equation takes the form: y+1/y=-2,5.

You have expressed the variable x in this equation using y.

You have expressed the variable x in this equation using y.