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9.(a)
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y = 2x + 1
Explanation: No work required! Simply plug in and write the answer.
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(b)
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x = 2
Explanation: This is the only possibility, since we have a straight line with
constant x.
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(c)
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x = 5
Explanation: Parallel to the y-axis
means vertical; 5 units to the left means that the line passes through the point
(5, 0). Again, no work is required; we simply write the answer by
inspection.
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(d)
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y = 3x 2
Explanation: m = Dy/Dx = (16 4)/(6 2) = 12/4 = 3. Use
point-slope form to write
y 4 = 3(x 2) by inspection. You can stop there, or you can use
algebraic simplification to get y =
3x 6 + 4 or y = 3x 2
as claimed.
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(e)
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y = ½x 2
Explanation: m = ½ (given), and
saying x-intercept of 4 means
that the point (4, 0) is a known point. Use point-slope form to write
y 0 = ½(x 4) by inspection. You can stop there, or you can use
algebraic simplification to get y
= ½x 2 as claimed.
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(f)
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y = 3x 7
Explanation: The slope of the first line is 3 by inspection, and the y-intercept of the second line is 7
by inspection. We can combine those facts to write y = 3x 7 in
slope-intercept form.
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(g)
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y = ½x 3
Explanation: Saying x-intercept of
6 means that (6, 0) is a point on the line. Similarly, saying y-intercept of 3 means that (0, 3)
is a point on the line. Since 2 points determine a line, we need only compute
a slope to be able to write the equation instantly. We get m = Dy/Dx = (3 0)/(0
6) = ½. Use everybodys favorite equation, y = mx
+ b, to get y = ½x 3
as claimed.
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10.(a)
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½ (since we can use algebra to rewrite equation as y = ½x + 5)
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(b)
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2 (by inspection)
Since ½ and 2 are opposite reciprocals, the lines are ^.
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11.
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Yes.
Explanation: The slope from (2, 4) to (5, 13) is 9/3 = 3.
Also, the slope from (5, 13) to (26, 76) is 63/21 = 3.
Since the slope is the same from (2, 4) to (5, 13) as it is from (5, 13) to
(26, 76), the second segment must be a continuation of the same straight
line.
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12.
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2/3 by inspection
Explanation: Think about rise/run, where rise is negative as we travel from
left to right. Or, if you prefer, think of Dy/Dx, where Dy is negative since y
is decreasing.
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13.
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29/4
Explanation: This is a multiple-step problem. Students often find problems like
this difficult, especially if they do not have the patience to stay focused
all the way through. Use backward chaining: What must we know in order to
find the x-intercept? The answer is
an equation of the line. Do we have enough information to find the equation
of the line? Yes, if we apply the method of problem 9d.
Therefore, we must first find an equation of the line. Continuing with backward
chaining, we realize that finding an equation of the line means that we must first
find the slope:
m = Dy/Dx = (7 3)/(5 (2)) = 4/7.
Now plug in to get equation in point-slope form: y 3 = 4/7 (x + 2).
When the question asks for the x-intercept,
what is it asking for? Answer: The place where the line crosses the x-axis. The coordinates of such a
point are (x, 0), wlog. We say x
since we dont know what x is, and
we say 0 since we do know what y is. Well, if the point (x, 0) is on the line, we must be able
to plug (x, 0) into the equation of
the line.
(Why? . . . because the point (x, 0) must satisfy the
linear equation if (x, 0) is on the
line.)
Lets do that. Take the point-slope equation from above: y 3 = 4/7 (x + 2).
Plug in x for x and 0 for y. We get
this result: 0 3 = 4/7 (x + 2).
Apply a little bit of Algebra I magic, and the answer turns out to be x = 29/4 as claimed.
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