77 F 224 CHAPTER 3. INTRODUCTION TO
GRAPHING Exercises In Exercises
1-6, set up a coordinate system on a sheet of
graph paper, then sketch the line through the
given point with the given slope. Label the line
with its equation in point-slope form. 1. m =
line is 5x + 2y = 6. Note that all the coefficients are integers and the terms are arranged
in the order Ax + By = C, with Answer: 3x 4y =
2 A 0. Point-Slope to Standard Form Lets do
an example where we have to put the pointslope form of a line in standar
both equations, it is a solution of the system.
Answer: (3, 4) Substitution method. The
substitution method involves these steps: 1.
Solve either equation for either variable. 2.
Substitute the result from step one into the
other equation. Solve the resul
option of which variable you choose to
eliminate, lets try Example 1 a second time,
this time eliminating y instead of x. You Try It!
EXAMPLE 2. Solve the following system of
equations. x + 2y = 5 (4.23) 2x y = 5 (4.24)
Solution: This time we focus on eli
comments are in order regarding the lines in
Figures 3.99 and 3.100. 1. The graph of x = 3 in
Figure 3.99, being a vertical line, has undefined
slope. Therefore, we cannot use either of the
formulae y = mx + b or y y0 = m(x x0) to
obtain the equation of t
ENTER key to produce the fractional equivalent
of the decimal content of the variable Y (see
Figure 4.35). Note that the fractional
equivalents for X and Y are 70/23 and 36/23,
precisely the same answers we got with the
substitution method above. Answer:
4 Again, note that (x, y) = (2.6, 1.4) does not
check exactly, but it is pretty close to being a
true statement. Because (x, y)=(2.6, 1.4) very
nearly makes both equations a true statement,
it seems that (x, y)=(2.6, 1.4) is a reasonable
approximation for
sides by 3. Thus, our system is equivalent to
the following two equations. 5 5 5 5 x y 2x +
3y = 6 y = 2 3 x + 4 Figure 4.36: 2x + 3y = 6
and y = 2 3x + 4 are parallel. No solution. y =
2 3 x + 2 (4.15) y = 2 3 x + 4 (4.16) These
lines have the same slope
line to determine the equation of the line. c)
Replace x and y in the equation found in part
(b) with K and F, respectively, then solve the
resulting equation for F. d) Use the result of
part (c) to determine the Fahrenheit
temperature of the object if th
ourselves some checking work by writing the
equation y 1 = 3 4 (x + 1) in the form y = 3 4
(x + 1) + 1 by adding 1 to both sides of the first
equation. Next, enter each equation as shown
in Figure 3.76, then change the WINDOW
setting as shown in Figure 3.
graph of the line 2x + 3y = 6. 5 5 5 5 x y 2 3
(0, 2) 2x + 3y = 6 Figure 4.9: Drawing the
graph of the line 2x + 3y = 6. To find the
solution of System 4.3, draw both lines on the
same coordinate system (see Figure 4.10).
Note how the lines appear to be p
of the actual solution. Then, in Sections 4.2
and 4.3, well show how to find the exact
solution. are plotted in Figure 4.6 and the line
2x + y = 4 is drawn through them. 10 10 10
10 x y (5, 0) (0, 3) 3x 5y = 15 Figure 4.5:
Drawing the graph of the line 3x
3.94, the graph crosses the x-axis three times.
Each of these crossing points is called an xintercept. Note that each of these x-intercepts
has a y-coordinate equal to zero. This leads to
the following rule. x-Intercepts. To find the xintercepts of the gr
by 5 using the distributive property. Enter y =
3+ 3 2x and y = 4 4 5x into the Y= menu of
the graphing calculator (see Figure 4.32). Press
the ZOOM button and select 6:ZStandard.
Press 2ND CALC to open the CALCULATE menu,
select 5:intersect, then press t
is 3x 4y = 2. Answer: 5x + 6y = 27 Horizontal
and Vertical Lines Here we keep an earlier
promise to address what happens to the
standard form Ax + By = C when either A = 0 or
B = 0. For example, the form 3x = 6, when
compared with the standard form Ax + B
6, or equivalently: 6x + 4y = 2 This doesnt
look like the same answer, but if we divide
both sides by 2, we do get the same result. 3x
2y = 1 This shows the importance of
requiring A 0 and reducing the coefficients
A, B, and C. It allows us to compare ou
Equation (4.8) is already solved for y.
Substitute equation (4.8) into equation (4.7).
This means we will substitute 3x1 for y in
equation (4.7). 2x 5y = 8 Equation (4.7). 2x
5(3x 1) = 8 Substitute 3x 1 for y in (4.7).
Now solve for x. 2x 15x +5= 8 Distr
curve as the First curve. Answer yes by
pressing the ENTER button. Having the
calculator ask First curve, Second curve,
when there are only two curves on the screen
may seem annoying. However, imagine the
situation when there are three or more curves
on t
drawn lines in Figure 3.84. This gives us
confidence that weve captured the correct
answer. Answer: y = 2x + 7 222 CHAPTER 3.
INTRODUCTION TO GRAPHING Figure 3.87:
Change the WINDOW parameters as shown.
Figure 3.88: 5:ZSquare produces two lines that
do lo
4y = 12 is 4/3 (see Figure 3.97). 5 5 5 5 x y
y = 4 x = 3 (3, 0) (0, 4) Figure 3.97: The
graph of 4x+3y = 12 has intercepts (3, 0) and
(0, 4) and slope 4/3. 5 5 5 5 x y y = 3 x =
4 (2, 2) Figure 3.98: The slope of the
perpendicular line is the negative re
name. The strategy is to somehow add the
equations of a system with the intent of
eliminating one of the unknown variables.
However, sometimes you need to do a little bit
more than simply add the equations. Lets look
at an example. You Try It! Solve the f
Finally, assuming a linear relationship between
the Celsius and Fahrenheit temperatures, draw
a line through these two points (see Figure
3.89). Calculate the slope of the line. m = C
F Slope formula. m = 100 0 212 32 Use the
points (32, 0) and (212, 100)
Ax + By = C Original equation. Ax + By Ax = C
Ax Subtract Ax from both sides. By = C Ax
Simplify. By B = C Ax B Divide both sides by B,
possible if B = 0. y = C B Ax B On the left,
simplify. On the right distribute the B. y = A B
x + C B Commutative prop
called the standard form of a line. SlopeIntercept to Standard Form Weve already
transformed a couple of equations in standard
form into slopeintercept form. Lets reverse
the process and place an equation in
slopeintercept form into standard form. 3.6.
ST
the same line. Parallel Lines Recall that slope is
a number that measures the steepness of the
line. If two lines are parallel (never intersect),
they have the same steepness. Parallel lines. If
two lines are parallel, they have the same
slope. You Try It
system of equations. He invented a method
(called Gaussian elimination) that is still used
today. Solving systems of equations has been a
subject of study in many other cultures. The
ancient Chinese text Jiuzhang Suanshu
(translated as Nine Chapters of Ma
solve the following system of Solve the
following system of equations: 2x 5y = 9 y =
2x 5 equations: 3x + 2y = 12 y = x + 1 (4.5)
Solution: To enter an equation in the Y= menu,
the equation must first be 254 CHAPTER 4.
SYSTEMS OF LINEAR EQUATIONS solved f
and R(4, 2) 23. P(2, 3), Q(1, 1), and R(3,
4) 24. P(4, 4), Q(1, 4), and R(4, 3) 3.5.
POINT-SLOPE FORM OF A LINE 225 25. Assume
that the relationship between an objects
velocity and its time is linear. At 3 seconds, the
objects velocity is 50 ft/s. At 14 s
247 To find the x-intercept, let y = 0. 3x + 2y =
12 3x + 2(0) = 12 3x = 12 x = 4 To find the yintercept, let x = 0. 3x + 2y = 12 3(0) + 2y = 12
2y = 12 y = 6 Hence, the x-intercept is (4, 0)
and the y-intercept is (0, 6). These intercepts
are plotted in