Question

Write an equation in slope-intercept form (y = mx + b) from the given information about a line.

Answer 1

Answers:
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1)   " y = - x – 5 " . 
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2)  " y = -2x + 2 "
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3)   y  =  " $\frac{7}{4}$  x  + $\frac{11}{4}$ "  .

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Explanation:
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1)  " y =  -x – 5 " .  

Note:  This equation is in the "slope-intercept form" ; that is:

  " y = mx + b" ; in which:  the slope, "m = -1 " ; 
                                          the y-intercept, "b = -5 " .
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1)  y = -x – 5 ;  


Note:  This equation is in the "slope-intercept form" ; that is:

  " y = mx + b" ; in which:  the slope, "m = -1 " ; 
                                          the y-intercept, "b = -5 " .
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2)   " y = -2x + 2 " .  

Note:  Given:  "(x₁, y₁)" ;  that is:  "(-2, 6)" ; in which:  "x₁ = -2" ; and:  "y₁ =6" ; 

And given the slope, "m", = -2 ; 

Use the formula:

" y – y₁ = m(x – x₁) " ; 

And substitute our known values:

" y – 6 = -2 [x – (-2)] " ;
 
  →  " y – 6 = -2 (x + 2) ; 

  →  " y – 6 = (-2*x) + (-2*2) ;
 
  →  " y – 6 = (-2*x) + (-2*2) ;

  →  " y – 6 =  -2x  + (-4) ;

  →  " y – 6 =  -2x – 4  ;

  →  Now, add "6" to EACH SIDE of the equation; to isolate "y" on the 
"left-hand side" of the equation; & write in "slope-intercept form" ;

  →  " y – 6 + 6 =  -2x – 4 + 6  ;

to get:

  →  " y = -2x + 2 " .  

Note:  This equation is in the "slope-intercept form" ; that is:

  " y = mx + b" ; in which:  the slope, "m = -2 " ; 
                                          the y-intercept, "b = 2 " .
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3)   " y  =  $\frac{7}{4}$  x  + $\frac{11}{4}$ "  .

 Given the points:  "(-1, 1)" ; and "(7, 15):

→  "(x₁, y₁)" ↔ "(-1, 1)" ; in which:  " x₁ = -1 " ;  " y₁ = 1 " ;

→ "(x₂ , y₂)" ↔  "(7, 15)" ; in which: " x₂ = 7 " ;  "y₂ = 15 " ; 
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Calculate the slope, "m" :

→   m = (y₂ – y₁) / (x₂ – x₁) ;
  =  (15 – 1) / [ 7 – (-1) ] = (15 – 1) / ( 7 + 1) ;

  =  $\frac{14}{8} = \frac{(14/2)}{(8/2)} = \frac{7}{4}$ ; 
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→   Now, use the formula:

→    " y – y₁ = m(x – x₁) " ; 

And substitute our known values:

→   " y – 1 = $\frac{7}{4}$ [x – (-1)] " ; 

→   " y – 1 = $\frac{7}{4}$ (x + 1) " ;


→   " y – 1 = $\frac{7}{4}$ x  +  $\frac{7}{4}$ " ;

→ Now, add "1" to EACH SIDE of the equation; to isolate "y" on the 
"left-hand side" of the equation; & write in "slope-intercept form" ;

→   " y – 1 + 1 = $\frac{7}{4}$ x  +  $\frac{7}{4}$  + 1 " ;

to get:

  →  " y  =  $\frac{7}{4}$  x  + $\frac{11}{4}$ "  .

And substitute our known values:

Note:  This equation is in the "slope-intercept form" ; that is:

  " y = mx + b" ; in which:  the slope, "m = $\frac{7}{4}$ " ; 

                                          the y-intercept, "b = $\frac{11}{4}$ " .
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