Linear Equations In Two Variables Worksheet (Page 3 of 7) in pdf

Linear Equations In Two Variables Worksheet Page 3

Systems of linear equations

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

variables, we could take two di↵erent linear equations in two variables and

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

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

x + y = 5 and x

y = 3.

If you have more than one linear equation, it’s called a system of linear

equations, so that

x + y = 5

y = 3

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: x and y.

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

the equations in the system.

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

linear equations in two variables

8x + 7y = 38

5y =

because x = 3 and y = 2 is a solution of 3x

5y =

1 and it is a solution of

8x + 7y = 38.

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 R

that lies on each of the

straight lines corresponding to the two linear equations.

Almost all of the time, two di↵erent 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

5y =

246

Linear Equations In Two Variables Worksheet Page 3