The table below shows a relationship between two sets of numbers, x and y.
Each ordered pair (x, y)
is a solution of the equation y = 4.2x.
The set of solutions of the
problem is called the solution
set. In this example, you would write the solution set as {(0.1,
0.42), (0.25, 1.05), (0.5, 2.1), (0.75, 3.15), (1.0, 4.2)}.
The equation y = 4.2x is
an example of an equation
in two variables.
Example
1. Which of
the ordered pairs (0, 1), (2, 3), (–1, 1), or (3, 5) are solutions of y =
2x – 1?
Alternative Solutions :
Make a table. Substitute the x and y
values of each ordered pair into the equation.
A true statement results when the ordered pairs (2, 3) and (3, 5) are substituted
into the equation. Therefore, ordered pairs (2, 3) and (3, 5) are solutions of
the equation y = 2x – 1.
Since the solutions of an equation in two variables are
ordered pairs, this type of equation describes a relation. The set of values of
x is the domain of the relation. The set of corresponding values of y
is the range of the relation.
Example
2.
Solve y
= 3x if the domain is {–2, –1, 0, 1, 3}. Graph the solution set.
Alternative Solutions :
Make a table. Substitute each value of x
into the equation to determine the corresponding values of y.
The solution set is {(–2, –6), (–1, –3), (0,
0), (1, 3), (3, 9)}.
Sometimes you can solve an equation for y before
substituting each domain value into the equation. This makes creating a table
of values easier.
Example
3.
Solve 4x
+ 2y = 8 if the domain is {–2, –1, 0, 1, 2}. Graph the solution set.
Alternative Solutions :
First, solve the equation for y in terms of x.
Next, substitute each value of x from the
domain to determine the corresponding values of y.
The solution set is {(–2, 8), (–1, 6), (0, 4), (1, 2), (2, 0)}.
You can also solve an equation for
given range values.
Example
4.
Find the
domain of y = 2x – 5 if the range is {–3, –1, 1, 3}.
Alternative Solutions :
Make a table. Substitute each value of y into the
equation. Then solve each equation to determine the corresponding values of x.
The domain is {1, 2, 3, 4}.
Sometimes variables other than x and y are used
in an equation. In this text, the values of the variable that comes first
alphabetically are from the domain.
Example
Geometry Link
5. The equation 2w + 2l = P
can be used to find the perimeter P
of a rectangle. Suppose a rectangle has a perimeter of 48 inches. Find the
possible dimensions of the rectangle given the domain values {3, 4, 8, 12}.
Alternative Solutions :
Assume
that the values of l come from the
domain. Therefore, the equation should be solved for w in terms of l.
Now,
substitute each value of l from the
domain into the equation to find the corresponding values of w.
The
dimensions of the rectangle could be (3, 21), (4, 20), (8, 16), or (12, 12).
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