Question

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

1. Slope = -1
y-intercept = -5

2. Slope = -2
y-intercept = Goes though points (-2, 6)

3. Slope = Goes though points (-1, 1)
Y-Intercept = (7, 15)

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 = "
`(7)/(4)` x +
`(11)/(4)` " .

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Step-by-step 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 =
`(7)/(4)` x +
`(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) ;

=
`(14)/(8) = ((14/2))/((8/2)) = (7)/(4)` ;
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→ Now, use the formula:

→ " y – y₁ = m(x – x₁) " ;

And substitute our known values:

→ " y – 1 =
`(7)/(4)` [x – (-1)] " ;

→ " y – 1 =
`(7)/(4)` (x + 1) " ;


→ " y – 1 =
`(7)/(4)` x +
`(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 =
`(7)/(4)` x +
`(7)/(4)` + 1 " ;

to get:

→ " y =
`(7)/(4)` x +
`(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 =
`(7)/(4)` " ;

the y-intercept, "b =
`(11)/(4)` " .
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