Linear Equations in Two Variables - University of Utah

Linear Equations in Two Variables

In this chapter, we'll use the geometry of lines to help us solve equations.

Linear equations in two variables

If , , and are real numbers (and if and are not both equal to 0) then

ab r

ab

ax + by = r is called a linear equation in two variables. (The "two variables"

are the and the .)

x

y

The numbers and are called the

of the equation + = .

ab

coe cients

ax by r

The number r is called the constant of the equation ax + by = r.

Examples. 10x 3y = 5 and 2x 4y = 7 are linear equations in two variables.

Solutions of equations

A solution of a linear equation in two variables ax+by = r is a specific point

in

R2

such

that

when

when

the

-coordinate x

of

the

point

is

multiplied

by

, a

and the -coordinate of the point is multiplied by , and those two numbers

y

b

are added together, the answer equals . (There are always infinitely many r

solutions to a linear equation in two variables.)

Example. Let's look at the equation 2 3 = 7. xy

Notice that x = 5 and y = 1 is a point in R2 that is a solution of this

equation because we can let = 5 and = 1 in the equation 2 3 = 7

x

y

xy

and then we'd have 2(5) 3(1) = 10 3 = 7.

The point = 8 and = 3 is also a solution of the equation 2 3 = 7

x

y

xy

since 2(8) 3(3) = 16 9 = 7.

The point x = 4 and y = 6 is not a solution of the equation 2x 3y = 7 because 2(4) 3(6) = 8 18 = 10, and 10 6= 7.

To get a geometric interpretation for what the set of solutions of 2 3 = 7 xy

looks like, we can add 3y, subtract 7, and divide by 3 to rewrite 2x 3y = 7

as 2 7 = . This is the equation of a line that has slope 2 and a -intercept

xy

y

33

3

of 7. In particular, the set of solutions to 2 3 = 7 is a straight line.

xy

(This3 is why it's called a linear equation.)

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Systems of linear equations

Rather than asking for the set of solutions of a single linear equation in two

variables, we could take two dierent linear equations in two variables and

ask for all those points that are solutions to of the linear equations. both

For example, the point = 4 and = 1 is a solution of both of the equations

x

y

x + y = 5 and x y = 3.

If you have more than one linear equation, it's called a

of linear

system

equations, so that

x+y = 5 =3

xy

is an example of a system of two linear equations in two variables. There are

two equations, and each equation has the same two variables: and .

xy

A

of equations is a point that is a solution of each of

solution of a system

the equations in the system.

Example. The point = 3 and = 2 is a solution of the system of two

x

y

linear equations in two variables

8 + 7 = 38 xy 3 5= 1 xy

because = 3 and = 2 is a solution of 3 5 =

x

y

xy

8 + 7 = 38. xy

1 it is a solution of and

Unique solutions

Geometrically, finding a solution of a system of two linear equations in two variables is the same problem as finding a point in R2 that lies on each of the straight lines corresponding to the two linear equations.

Almost all of the time, two dierent lines will intersect in a single point, so in these cases, there will only be one point that is a solution to both equations. Such a point is called the unique solution of the system of linear equations.

Example. Let's take a second look at the system of equations

8x + 7y = 38 3 5= 1 xy

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