Question

Write an equation for the line passing through the given point and having the given slope.

Answer 1

Answer:  The equation, in "slope-intercept form" ; is:

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                   →   " y  =   x  +  4  " .

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Step-by-step explanation:

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Use the formula for linear equations;  written in "point-slope format" ;

which is:

      y −  y₁  = m( x −  x₁ )  ;

We are given the slope, "m" ;  has a value of:  "1 " ;  

           that is;  " m = 2 " .

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  We are given the coordinates to 1 (one) point on the line; in which the coordinates are in the form of :

    " ( x₁ , y₁ ) " ;  

→   that given point is:  "(10, 6)" ;  

            in which:  x₁ =  10 ;

                             y₁  =  6 .

→ Given:  The slope, "m" equals "1" ;  ________________________________________________

Let's plug our known values into the formula:

    →  "  y  −  y₁  = m( x − x₁ ) "  ;

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    →   As follows:

    →   " y  −  10  = 1(x − 6) ;

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Now, focus on the "right-hand side of the equation" ;

    →    1(x  −  6) = ? ;   Simplify.

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Note the "distributive property" of multiplication:

            →  a(b + c) =  ab   +   ac ;

As such:  " 1(x  −  6) =  (1*x)  +  (1 * -6) " ;

                             =  1x   + (-6) ;

                             =   x  −  6 ;

[Note that:  " 1 x = 1 * x = x " ;

[Note that   " + (-6) "  =  " ( " − 6 " ) .] ;  

                 →  {since:  "Adding a negative" is the same as:

                                  "subtracting a positive."} ;

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          Now, let us bring down the "left-hand side of the equation" ; &

rewrite the entire equation; as follows:

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                           →   "  y  −  10  =   x  −  6 " ;  

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Note:  We want to rewrite the equation in "slope-intercept form" ;

           that is;  " y = mx + b "  ;

                 in which:  "y" ; stands alone as a single variable on the "left-hand side" of the equation;  with "no coefficients" [except for the "implied coefficient" of " 1 "} ;

                                 "m" is the coefficient of "x" ;

                                 and the "slope of the line" ;  

          Note that "m" may be a "fraction or decimal" ; and may be "positive or negative.

            If the slope is "1" ; (that is "1 over 1" ; or: "$\frac1}{1}$" ;

then,  " m = 1 " ;  and we can write " 1x " as simply "x" ; since the implied coefficient is "1" ;  

               →  since " 1" , multiplied by any value {in our case, any value for "x"} , equals that same value.

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       "b"  refers to the "y-intercept" of the graph of the equation;  

           that is; the "y-value" of the point at which the graphed line of the equation crosses the "y-axis" ;  

           that is, the "y-value" of the coordinates of the point at which the graphed line of the equation crosses the "y-axis" ;

           that is, the ["y-value" of the]  y-intercept" .  

Note that the value of "b" may be positive or negative, and may be a decimal or fraction.

  If the value for "b" is negative, the equation can be written in the form:

          " y = mx - b " ;

    {since:  " y = mx + (-b) "  is a bit tedious .}

          If the y-intercept is "0" ; (i.e. the line crosses the y-axis at the origin, at point:  " (0,0) " ;  

then we simply write the equation as:  "y = mx " ;  

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So;  we have:              →   " y  − 10  =  x  −  6 " ;  

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       →   We want to rewrite our equation in slope-intercept form,

that is;  " y = mx + b " ;  as explained above.

 

We can add "10" to each side of the equation ; to isolation the "y" on the "left-hand side" of the equation:

           →  " y  −  10  + 10  =  x  − 6  +  10 " ;

to get:

           →  " y  =  x  +  4 " ;

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          →  which is our answer.

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Note:  This answer:  " y = x + 4 " ;

                  →  is written in the "slope-intercept format";  

                     →   " y = mx +  b "  ;

  in which:  "y" is isolated as a single variable on the "left-hand side of the equation" ;

                   The slope of the equation is "1" ; or an implied value of "1" ;

    that is;   " m  = 1 " ;  

                   "b  =  4 " ;  

         →  {that is;  the "y-value" of the  "y-intercept" —  "(0, 4)" — of the graph of the equation is:  "4 ".} .

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Hope this answer is helpful!

          Best wishes to you in your academic pursuits

               —  and within the "Brainly" community!

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