3.2
Connecting y 5 mx 1 b and ax 1 by 5 c
Goals
• Connect the two forms of linear equations
y = mx + b and ax + by = c
• Practice working with algebraic equations
Launch 3.2
Going Further
Students needing an extra challenge might consider
the inverse of the question in Question C.
• If you have an equation in y = mx + b form,
how can you predict what a, b and c will be in
the equivalent standard form?
Because there are infinitely many equivalent
equations in standard form, answers will vary.
The simplest answer is that a = m,
b standard form = -1, and c = -b slope-intercept form.
This is based on the solution:
Use the Getting Ready to help launch the problem.
In the last problem, students developed the idea
that the graph of the solutions of the equation
ax + by = c is a straight line. They may already
see a connection with equations of the form
y = mx + b. However, students may have a sense
that these feel different: in equations of the form
y = mx + b, the functional relationship is explicit.
y clearly depends on x. In equations of the form
ax + by = c, the functional relationship is implicit.
The variable y can still be said to depend on x, but
not in such an obvious way; x might just as easily
depend on y. This problem offers students a chance
to make sure that indeed these two kinds of
equations describe the same kind of graph.
You could launch this problem by asking what
the graphs of y = mx + b and ax + by = c look
like, and then asking the question from the text:
Ask students whether they now think that every
equation of the form ax + by = c can be written in
an equivalent y = mx + b form. Ask what it means
that the two equations are equivalent. Focus on
Question A for a moment, and ask if the students in
the problem got an equation equivalent to the
original. Invite students in the class to try to guess
what the students in the problem were thinking.
Suggested Question
Suggested Questions
You might work on Question A as a whole class
so that the students see that the linear equation
ax + by = c can be written in y = mx + b form
and to model the kind of reasoning expected in the
remaining problem parts. In Question A, students
are asked to analyze someone else’s reasoning.
Then in Questions B through D they are to apply
what they’ve observed to some new problems on
their own.
Explore 3.2
As you circulate, listen to students’ reasoning
about why these algebraic manipulations work or
do not work.
• Why would these students be confused?
• How did they get tripped up?
• If you were their teacher, how would you help
3
always be written in equivalent y = mx + b
form?
Summarize 3.2
I N V E S T I G AT I O N
• Can linear equations such as ax + by = c
y = mx + b
-mx + y = b
mx - y =-b
them make sense of this algebra?
One thing that may be confusing is that the
parameter b shows up in both forms of the
equation. As students begin to work on moving
between the two equations, make sure that
students understand that the b in ax + by = c is
not the y-intercept of the graph—it’s just an
unfortunate notational coincidence.
Finally, it is not the case that all equations of
form ax + by = c can be written in y = mx + b
form. In the special case where b standard form = 0,
we are left with ax = c, which describes a vertical
line. Vertical lines have no slope, so there can be no
slope-intercept form of the equation. For all other
lines, though, we may write equations in both forms.
Investigation 3
Equations With Two or More Variables
61
62
The Shapes of Algebra
At a Glance
3.2
Connecting y 5 mx 1 b and ax 1 by 5 c
PACING 1 day
Mathematical Goals
• Connect the two forms of linear equations y = mx + b and ax + by = c
• Practice working with algebraic equations
Launch
Use the Getting Ready to help launch the problem. You could launch this
problem by asking what the graphs of y = mx + b and ax + by = c look
like, and then asking the question from the text:
• Can linear equations such as ax + by = c always be written in
equivalent y = mx + b form?
Materials
•
Transparencies 3.2A,
3.2B
Vocabulary
•
You might work on Question A as a whole class so the students see that
the linear equation ax + by = c can be written in y = mx + b form and to
model the kind of reasoning expected in the remaining problem parts. In
Question A, students are asked to analyze someone else’s reasoning. Then
in Questions B through D, they are to apply what they’ve observed to some
new problems on their own.
linear equation in
standard form
Explore
As you circulate, listen to students’ reasoning about why these algebraic
manipulations work or do not work.
Going Further: Students needing an extra challenge might consider the
inverse of the question in Question C.
• If you have an equation in y = mx + b form, how can you predict what
a, b, and c will be in the equivalent standard form?
Because there are infinitely many equivalent equations in standard form,
answers will vary. The simplest answer is that a = m, b standard form = –1,
and c = –b slope-intercept form.
Summarize
Ask students whether they now think that every equation of the form
ax + by = c can be written in an equivalent y = mx + b form. Ask what it
means that the two equations are equivalent. Focus on Question A for a
moment, and ask if the students in the problem got an equation equivalent
to the original.
Materials
•
Student notebooks
Vocabulary
•
slope-intercept form
• Why would these students be confused?
• How did they get tripped up?
Investigation 3
Equations With Two or More Variables
63
ACE Assignment Guide
for Problem 3.2
2. Two equations are equivalent if they have
the same solutions.
Core 15–27, 36–48
Other 49, 50, 63, and unassigned choices from
2. y = -2x + 9.
previous problems
3. y = -2x – 3.
Adapted For suggestions about adapting ACE
4. y = 2x – 2.
5. y = -x + 2.5.
exercises, see the CMP Special Needs Handbook.
Connecting to Prior Units 51, 57–62: Moving
Straight Ahead; 52: Say It With Symbols; 53, 56:
Filling and Wrapping; 54, 55: Frogs, Fleas, and
Painted Cubes
A. 1. Jared: Yes; Jared used properties of
equality: he subtracted 12x from both sides
of the equation in the first step, which
yielded an equivalent equation. Then he
divided both sides of the equation by 3,
continuing to maintain an equivalent
equation. Dividing the right side -12x + 9
1
by 3 is the same as multiplying it by 3, so
1
(-12x + 9) 4 3 = (-12x + 9) 3 = -4x + 3.
Molly: No; she needed to divide both sides
of the equation by 3. When she divided 9 –
12x by 3, she did not use the Distributive
Property and forgot to divide -12x by 3.
Ali: Yes; as in A, he maintained equivalent
equations throughout his work. First he
divided both sides of the equation by 3 and
then subtracted 4x from either side.
Mia: No; 2.4x – 3 is not equivalent to
3 2 4x.
The Shapes of Algebra
1
1
6. y = -2x + 60.
a
c
C. The slope is -b. The y-intercept is b and the
c
x-intercept is a. Since by = -ax + c,
a
c
y = -bx + b.
Answers to Problem 3.2
64
B. 1. y = x – 4.
D. Answers will vary, as there are multiple
equivalent forms for all equations in standard
form. The answers given below use the
smallest (in absolute value) whole-number
coefficients.
1. 3x + y = 5
2. 8x - 12y = -3
3. x - 2y =-3
4. 4x - 2y = 1
5. x - 4y = -12
6. 2x - y = 6