ADVERTISEMENT

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 a, b, and r are real numbers (and if a and b are not both equal to 0) then

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

are the x and the y.)

The numbers a and b are called the coe cients of the equation 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 R

2

such that when when the x-coordinate of the point is multiplied by a,

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

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

solutions to a linear equation in two variables.)

Example. Let’s look at the equation 2x

3y = 7.

Notice that x = 5 and y = 1 is a point in R

2

that is a solution of this

equation because we can let x = 5 and y = 1 in the equation 2x

3y = 7

and then we’d have 2(5)

3(1) = 10

3 = 7.

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

3y = 7

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 = 7.

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

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

3y = 7

2

7

2

as

x

= y. This is the equation of a line that has slope

and a y-intercept

3

3

3

7

of

. In particular, the set of solutions to 2x

3y = 7 is a straight line.

3

(This is why it’s called a linear equation.)

244

ADVERTISEMENT

0 votes

Parent category: Education