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Section 5-5
Standard Form and Intercepts
Alg. I
An equation is “linear” if both variables occur as first powers only,
there are no products of variables, and no variable is in a denominator.
The most basic of linear equations actually only shows one of the
two variables but not both. These equations produce either horizontal
or vertical lines (not diagonal!)
EXAMPLES:
B) y  2
Graph A) x  3
Solutions:
The solution set for A) will ALWAYS have x = 3, regardless of what
you choose for a y value. The solution set for B) will ALWAYS be – 2,
regardless of what you choose for an x value:
Listing of the “Table of Values” (or “T-chart”) for each:
A) x
3
3
3
3
y
0
2
-1
½
y
(3,2)
(3, 1 2 )
(3,0)
y
B)
x
(3,1)
For constants a and b:
 The graph of y = b is a horizontal line.
 The graph of x = a is a vertical line.
Solutions:
y
A)
B)
(0,2)
(4,2)
x
5
0
-4
¼
x
( 4 , 2 )
1
(5,2)
y
x3
x
x
y  2
YOUR Notes:
y
-2
-2
-2
-2
Section 5-5 (cont.)
Standard Form and Intercepts
Alg. I
QUESTION: What’s the difference between the list of answers
(the graphs) of these four equations?:
a) 2 x  12 y  15 b)
2
3
x  4 y  5 c)  30  4 x  24 y d)  5  4 y  23 x
ANSWER: Nothing. For continuity, linear equations are listed in
STANDARD FORM. This form is Ax  By  C , where
1) The x and y variables are alphabetical on the same side of the
equal sign.
2) C is a constant, and
3) A and B are integers (not fractions!).
EXAMPLES: Rewrite in “Standard Form”:
A)  4 x  y  1
1
3
B)
Solutions:
A)
y  x  2
C) y  x
B)
 4x
 y 1
  y  y 
 4x  y 
1
C)
y  x  2
 x   x 
x  13 y 
2
3x  13 y  2 3
3x  y  6
1
3
y x
x
 x
x y0
…Open textbooks to page 323 and try “Got It?” problems 3 and 4
y
x4
#3 a
#3 b
y
x  1
P. 324 #4
x
y  2   13 ( x  6)
x
y  2   13 x  2
 13 x     13 x 
1
3
y
#3 c
#3 d
y 1
x
y0
(The x-axis!)
y
x y2
 2
3 13 x  y 
x  3y

2
2
0 3
0
Section 5-5 (cont.)
QUICK GRAPHS USING INTERCEPTS
You can quickly sketch the graph of a linear equation by plotting its
“intercepts”. The graph shows the relationships between variables.
For ‘linear’ equations:
a) The graph of any linear equation in form Ax  By = C, is a line.
b) Two points are the minimum needed to determine a line.
Recall:
X-Intercept: (?, 0) - Location where a graph crosses over the x-axis
from left to right on a coordinate plane.
Y-Intercept: (0, ?) – Location where a graph crosses over the y-axis
from left to right on a coordinate plane.
Example: Sketch the graph of
2x + 3y = 12 by using intercepts.
Solution. There’s no need to
solve for y. Just evaluate each
by placing 0 for each variable:
2(0)  3 y  12
x y
0 ?
? 0
Examples: Sketch the graphs
of 3 x  5 y  15 and  2 x  3 y  9
by using intercepts.
Solutions:
x y
0 ?
? 0
0  3 y  12
3y
3
 123
3(0)  5 y  15
 2(0)  3 y  9
0  5 y  15
0  3y  9
y4
5 y
5
2 x  3(0)  12
2x  0  12
2x
2
x y
0 ?
? 0
x
0
6
 122
x6
y
4
0

3y
3
15
5
3 x  5(0)  15

x 5
9
3
y 3
y  3
x y
0 -3
5 0

x y
0 3
-4.5 0
 2x  3y  9

x  4.5
(0,4)
(5,0)
(6,0)
(0,3)
2 x  3 y  12
3 x  5 y  15
(4.5,0)
 2x  3y  9
(0,3)