Linear Equations In Two Variables Worksheet
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7Linear 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.
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(This is why it’s called a linear equation.)
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