The relation at the right is a solution set for the equation y = x + 2. If you graph the relation, the points lie on a straight line, and the graph of the line includes all possible solutions of the equation of the graph.
The line contains an infinite number of points. The ordered
pair for every point on the line is a solution of the equation y = x +
2. An equation whose graph is a straight line is called a linear equation.
Linear equations can contain one or two variables. However,
not all equations in one or two variables are linear equations.
Linear equations Nonlinear
equations
3x
+ 4y = 6 2x + 4y
= 3z2
y = –2 2x
+ 2xy = 3
8b
= –2a – 5 2x2 = 7
Example
Determine whether each equation is a linear equation.
Explain. If an equation is linear, identify A, B, and C when
written in standard form.
1. 3x = 5 + y
Alternative Solutions :
First,
rewrite the equation so that both variables are on the same side of the
equation.
3x = 5 + y
3x – y
= 5 Subtract y from each side.
The
equation is now in the form Ax + By + C,
where A = 3, B = –1, and C = 5.
Therefore, this is a linear equation.
2.
–2x + 3xy
= 6
Alternative Solutions :
Since
the term 3xy has two variables, the equation cannot be written in the form Ax + By = C. So, this is not a linear
equation.
3.
x = –2
Alternative Solutions :
This
equation can be written as x + 0y = –2. Therefore, it is a linear equation
in the form Ax + By = C, where A = 1, B = 0, and C = –2.
To graph a linear equation, make a table of ordered pairs
that are solutions. Graph the ordered pairs and connect the points with a line.
Note that the domain of the equation and its graph includes the values of x and
the range includes the values of y.
Example
Graph each equation.
4. y = 2x
– 3
Alternative Solutions :
Select several values for
the domain and make a table. Then graph the ordered pairs and connect them to
draw the line.
Usually, values of x
such as 0 and integers near 0 are chosen so that calculations are simple.
5.
2x +
4y = 8
Alternative Solutions :
In order to find values for y more easily,
solve the equation for y.
Now make a table and draw the graph.
Sometimes, the graph of an equation is a vertical or
horizontal line.
Example
6. Graph y = 3. Describe the
graph.
Alternative Solutions :
In
standard form, this equation is written as 0x
+ y = 3. So, for any value of x, y
= 3. For example, if x = 0, y = 3; if x = 1, y = 3; if x = 3, y = 3. By graphing the ordered pairs (0, 3), (1, 3), and (3, 3),
you find that the graph of y = 3 is horizontal
line.
Recall that the domain of a linear function is all of
the x-coordinates of the ordered pairs for points on the line. The range
of a linear function is all of the y-coordinates of the ordered pairs
for points on the line. Since a line extends forever in both directions, if it
is not vertical and not horizontal, the domain and the range are both the set
of all real numbers.
Example
7. Graph y = 2x.
Alternative Solutions :
Make a
table and draw the graph.
Write y = 2x in standard form.
y = 2x Original
equation
0 = 2x – y Subtract y from each side.
2x – y = 0 Symmetric
Property of Equality
In any
equation where C = 0, the graph
passes through the origin.
Example
Earth
Science
8. Old Faithful is a famous geyser. The equation y = 14x + 27 can be used
to predict its eruptions, where y is
the time until the next eruption (in minutes) and x is the length of the eruption (in minutes). Suppose Old Faithful
erupts at 9:46 A.M. for 3.4 minutes. At about what time will the next eruption
occur?
Alternative Solutions :
Explore
You need to find the time of the
next eruption.
Plan Use the graph of y = 14x + 27 to solve the problem.
Solve The graph shows that if an eruption
lasts 3.4 minutes,
the next eruption will occur in about 75 minutes
The next eruption should take place at 11:01 A.M.
Examine
You can verify the
answer by using the equation.
y = 14x + 27 Original
equation
y ≟ 14(3.4) + 27 Replace x with 3.4.
y = 74.6 Simplify.
The answer makes sense because 74.6 minutes added to
9:46 is about 11:01.
Sumber
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