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Important aspects of linear equations are transforming equations, understanding linear patterns, discovering the slope, y-intercept and points.
There are many reasons you may need to create a new linear equation.
You may need to transform one format to the another format to solve a problem.
Writing equations can assist in understanding the pattern of data from a table.
WRITING EQUATIONS FROM VARIOUS ASPECTS
Writing equations by converting equations with equations, points and tables
CONVERTING EQUATIONS
Converting standard and slope-intercepts forms
STANDARD FORM TO SLOPE-INTERCEPT FORM
Explanation of converting standard form to slope-intercept form
Standard Form
Ax + By = C
Ax + By + C = 0 --standard form
Ax + By + C - C = 0 - C
Ax + By + 0 = -C
Ax - Ax + By = -Ax - C
0 + By = -Ax - C By = -Ax - C
B B B
y = -Ax - C
B B
where,
m = -Ax
B
and
y-intercept = b =
y = mx + b --slope-intercept
C
B
Standard Form to Slope-Intercept Form
Examples of converting formats
2x - y = 3
2x - 2x - y = 3 - 2x
0 - y = -2x + 3
-y = -2x + 3
y = 2x - 3
-1 -1 -1
x - x -2y = -x - 8
0 - 2y = -x - 8
-2y = -x - 8
y = 1x + 4
-2 -2 -2
x - 2y = -8
2
3x - 9y = -18
-9 -9 -9
3x - 3x - 9y = -3x - 18
0 - 9y = -3x -18
-9y = -3x -18
y = 3x + 2
9
1
change in y = A
change in x = B
1
Slope-Intercept form to Standard Form
Explanation of converting slope-intercept form to standard form
Slope-Intercept
y = mx + b
m =
y - y
2
x - x
2
=
when
-Ax = 0
B
then,
y-intercept = b = C
B
y = mx + b OR
y = -Ax - C
(B)y = -Ax (B) + C (B)
By = -Ax + C --updated expression
Ax + By = -Ax + Ax + C --subtract Ax from both sides
Ax + By = 0 + C
Ax + By = C --standard form
B B
B B
--multiply B by each side
Slope-Intercept Form to Standard Form
Examples of converting formats
(3)y = -7x (3) - 12 (3)
3y = -7x - 36
7x + 3y = -7x + 7x - 36
7x + 3y = 0 - 36
7x + 3y = -36
y = -7x - 12
3
3
(5)y = -3x (5) + 5(5)
5y = -3x + 25
3x + 5y = -3x + 3x + 25
3x + 5y = 0 + 25
3x + 5y = 25
y = -3x + 5
5
5
y = 4x + 6
-4x + y = 4x - 4x + 6
-4x + y = 0 + 6
-4x + y = 6
CONVERTING WITH TWO POINTS
Converting two coordiante points into slope-intercept format
TWO POINTS: POINT-SLOPE TO SLOPE-INTERCEPT FORM
Explanation and example of converting two points into slope-intercept form utilizing point-slope form
2 POINTS
A line passing through points (-2, 5), (2, 1)
1
2
2
1
x = -2
x = 2
y = 5
y = 1
Discover the slope
m =
x - x
2
y - y
2
1
1
=
1 - 5
2 - (-2)
-1
-4
2 + 2
=
-4
4
=
=
Convert utilizing the point-slope format
(y - y ) =
1
m
(x - x )
1
y -(5) = -1(x - 2) --substitute slope, x-value & y-value
y - 5 + 5 = -1x - 2 + 5
y + 0 = -1x + 3
y = -1x + 3 --slope-intercept
where,
1
1
x = 2
y = 5
m = -1
CONVERTING WITH ONE POINT
Converting coordinate points into slope-intercept form
ONE POINT: POINTS INTO SLOPE-INTERCEPT FORM
Explanation and example of converting one point into slope-intercept form utilizing point-slope form
1 Point & Slope
A line passing through points (2, 3), with a slope of 4
1
1
x = 2
y = 3
m = 4
where,
Convert utilizing the point-slope format
1
(y - y ) =
1
m
(x - x )
y -(3) = 4(x - 2) --substitute slope, x-value & y-value
y - 3 = 4x - 8
y - 3 + 3 = 4x - 8 + 3 --add 3 to both sides
y + 0 = 4x - 5
y = 4x - 5 --slope-intercept
CONVERTING FROM TABLES
Converting coordinate tables to slope-intercept
Solve the Equation
x
2x + 3
-3
-2
-1
0
2(-3) + 3
2(-2) + 3
2(-1) + 3
2(0) + 3
y
-3
-1
1
3
=
=
=
=
Discover the slope from the table
x
-3
-2
-1
0
+1
+1
+1
y
-3
-1
1
3
+2
+2
+2
change in x = 1
change in y = 2
Rate of change or slope
m =
change in y
change in x
=
2
1
= 2
Convert utilizing the point-slope format
Choose a point coordinate ( 1, 5) from the table
1
1
x = 1
y = 5
m = 2
where,
(y - y ) =
1
m
(x - x )
1
(y - 5) = 2(x - 1) --substitute x-value, y-value & slope
y - 5 = 2x - 2
y - 5 + 5 = 2x - 2 + 5 --add 5 to both sides
y + 0 = 2x + 3
y = 2x + 3 --slope-intercept
m
(x - )
x
1
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