3) Give 10 examples of Linear Equations in two Variables. Write at least three solutions of each.
Answers
Answer:
Step-by-step explanation:
. 10x 3y = 5 and 2x 4y = 7
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 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 R2 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 6= 7.
To get a geometric interpretation for what the set of solutions of 2x3y = 7
looks like, we can add 3y, subtract 7, and divide by 3 to rewrite 2x 3y = 7
as 2
3x 7
3 = y. This is the equation of a line that has slope 2
3 and a y-intercept
of 7
3. In particular, the set of solutions to 2x 3y = 7 is a straight line.
(This is why it’s called a linear equation.)