3 = 3 O True O False
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Given the the equation: y − 7 = x + 4 ;
We should rewrite the equation in "slope-intercept form" ; that is:
" y = mx + b " ;
______________________
in which: "y" remains an isolated variable on the "left-hand side" of the equation (with "no coefficients"—except the "implied coefficient of "1"
{since "1" , multiplied by any value, results in that same value} ;
"m" refers to the "slope" of the line;
and "m" is the coefficient of "x" ;
"b" refers to the "y-intercept" of the graphed (linear) equation; or more precisely, the value of "y" of the coordinate of the [ point on the graphed line of the equation—where the line crossed the "y-axis" ] ; that is; the "y-coordinate of the "y-intercept" ; that is, the value of "y" when "x = 0" ;
______________________________________________________
{Note that: "m" (the slope) may be a negative value; & that: "y" may be a negative value; as regard to the "slope-intercept form" equation:
that is:
"y = mx + b" . }.
______________________________________________________
So; given the equation:
______________________________________________________
" y − 7 = x + 4 " ;
______________________________________________________
Add "7" to each side of the equation;
to isolate "y" on one side of the equation;
→ {specifically, the "left-hand-side" of the equation} ;
and to rewrite the equation in "slope-intercept form" ; that is;
→ to rewrite the equation in the format:
" y = mx + b " ;
_______________________________________________________
As such: "y − 7 = x + 4 " ;
_______________________________________________________
→ Add "7" to each side of the equation:
_______________________________________________________
→ y − 7 + 7 = x + 4 + 7 ;
to get:
_______________________________________________________
→ y = x + 11 ;
_______________________________________________________
Now, let us graph the equation:
_______________________________________________________
Let us make a table:
_______________________________________________________
x ║ (x + 11) ║ y ║
_______________________________________________________
-20 ║ (-20 + 11) ║ -9 ║
-19 ║ (-19 + 11) ║ -8 ║
-18 ║ (-18 + 11) ║ -7 ║
-17 ║ (-17 + 11) ║ -6 ║
-16 ║ (-16 + 11) ║ -5 ║
-15 ║ (-15 + 11) ║ -4 ║
-14 ║ (-14 + 11) ║ -3 ║
-13 ║ (-13 + 11) ║ -2 ║
-12 ║ (-12 + 11) ║ -1 ║
-11 ║ (-11 + 11) ║ 0 ║
-10 ║ (-10+ 11) ║ 1 ║
-9 ║ (-9 + 11) ║ 2 ║
-8 ║ (-8 + 11) ║ 3 ║
-7 ║ (-7 + 11) ║ 4 ║
-6 ║ (-6 + 11) ║ 5 ║
-5 ║ (-5 + 11) ║ 6 ║
-4 ║ (-4 + 11) ║ 7 ║
-3 ║ (-3 + 11) ║ 8 ║
-2 ║ (-2 + 11) ║ 9 ║
-1 ║ (-1 + 11) ║ 10 ║
0 ║ (0 + 11) ║ 11 ║
1 ║ (1 + 11) ║ 12 ║
2 ║ (2 + 11) ║ 13 ║
3 ║ (3 + 11) ║ 14 ║
4 ║ (4 + 11) ║ 15 ║
5 ║ (5 + 11) ║ 16 ║
_________________________________________________
Now, we plot these points on a graph, and draw a line through these points; the slope is "1" ; since the equation is:
" y = x + 11 " ; Note: The slope, "m", is "1", since there is an "implied coefficient of "1" ; i.e. " y = 1x + 11 " .
___________________________________
Graph of: "y = x + 11 " ;
_____________________________________
We should rewrite the equation in "slope-intercept form" ; that is:
" y = mx + b " ;
______________________
in which: "y" remains an isolated variable on the "left-hand side" of the equation (with "no coefficients"—except the "implied coefficient of "1"
{since "1" , multiplied by any value, results in that same value} ;
"m" refers to the "slope" of the line;
and "m" is the coefficient of "x" ;
"b" refers to the "y-intercept" of the graphed (linear) equation; or more precisely, the value of "y" of the coordinate of the [ point on the graphed line of the equation—where the line crossed the "y-axis" ] ; that is; the "y-coordinate of the "y-intercept" ; that is, the value of "y" when "x = 0" ;
______________________________________________________
{Note that: "m" (the slope) may be a negative value; & that: "y" may be a negative value; as regard to the "slope-intercept form" equation:
that is:
"y = mx + b" . }.
______________________________________________________
So; given the equation:
______________________________________________________
" y − 7 = x + 4 " ;
______________________________________________________
Add "7" to each side of the equation;
to isolate "y" on one side of the equation;
→ {specifically, the "left-hand-side" of the equation} ;
and to rewrite the equation in "slope-intercept form" ; that is;
→ to rewrite the equation in the format:
" y = mx + b " ;
_______________________________________________________
As such: "y − 7 = x + 4 " ;
_______________________________________________________
→ Add "7" to each side of the equation:
_______________________________________________________
→ y − 7 + 7 = x + 4 + 7 ;
to get:
_______________________________________________________
→ y = x + 11 ;
_______________________________________________________
Now, let us graph the equation:
_______________________________________________________
Let us make a table:
_______________________________________________________
x ║ (x + 11) ║ y ║
_______________________________________________________
-20 ║ (-20 + 11) ║ -9 ║
-19 ║ (-19 + 11) ║ -8 ║
-18 ║ (-18 + 11) ║ -7 ║
-17 ║ (-17 + 11) ║ -6 ║
-16 ║ (-16 + 11) ║ -5 ║
-15 ║ (-15 + 11) ║ -4 ║
-14 ║ (-14 + 11) ║ -3 ║
-13 ║ (-13 + 11) ║ -2 ║
-12 ║ (-12 + 11) ║ -1 ║
-11 ║ (-11 + 11) ║ 0 ║
-10 ║ (-10+ 11) ║ 1 ║
-9 ║ (-9 + 11) ║ 2 ║
-8 ║ (-8 + 11) ║ 3 ║
-7 ║ (-7 + 11) ║ 4 ║
-6 ║ (-6 + 11) ║ 5 ║
-5 ║ (-5 + 11) ║ 6 ║
-4 ║ (-4 + 11) ║ 7 ║
-3 ║ (-3 + 11) ║ 8 ║
-2 ║ (-2 + 11) ║ 9 ║
-1 ║ (-1 + 11) ║ 10 ║
0 ║ (0 + 11) ║ 11 ║
1 ║ (1 + 11) ║ 12 ║
2 ║ (2 + 11) ║ 13 ║
3 ║ (3 + 11) ║ 14 ║
4 ║ (4 + 11) ║ 15 ║
5 ║ (5 + 11) ║ 16 ║
_________________________________________________
Now, we plot these points on a graph, and draw a line through these points; the slope is "1" ; since the equation is:
" y = x + 11 " ; Note: The slope, "m", is "1", since there is an "implied coefficient of "1" ; i.e. " y = 1x + 11 " .
___________________________________
Graph of: "y = x + 11 " ;
_____________________________________