Step 1: Find the slope between the two points:

\(\displaystyle \frac {y_2-y_1}{x_2-x_1}=\frac {8-4}{7-2}=\frac {4}{5}\)

Step 2: Write the slope-intercept form:

\(\displaystyle y=\frac {4}{5}x+b\)

Step 3. Find b. Plug in (x,y) from one of the points:

\(\displaystyle 4=\frac {4}{5}(2)+b\)

\(\displaystyle 4=\frac {8}{5}+b\)

\(\displaystyle 4-\frac {8}{5}=b\)

\(\displaystyle \frac {20-8}{5}=b\)

\(\displaystyle \frac {12}{5}=b\)
Step 4: Write out the full equation:

\(\displaystyle y=\frac {4}{5}x+\frac {12}{5}\)

To find the equation of the line, first find the slope using the formula:

\(\displaystyle m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

The points that the line passes through are \(\displaystyle (3,4)\) and \(\displaystyle (1,-2)\).

\(\displaystyle m = \frac{-2-4}{1-3}\)

\(\displaystyle m= \frac{-6}{-2}\)

\(\displaystyle m = 3\)

Then pick one set of points and place in the form   \(\displaystyle y = mx+b\). Either set of points will give you the same equation.  Points \(\displaystyle (3,4)\) were used.

\(\displaystyle 4 = 3\times 3 + b\)

\(\displaystyle 4 = 9 +b\)

Subtract  \(\displaystyle 9\)  from both sides of the equation.

\(\displaystyle 4-9 = 9-9 +b\)

\(\displaystyle -5 = b\)

The equation of the line in slope-intercept form or    is

 \(\displaystyle y= 3x-5\)

To find the equation of a line that passes through \(\displaystyle (-1,3)\) and \(\displaystyle (-2,5)\), first find the slope using this formula:

\(\displaystyle m = \frac{y_{2} -y_{1}}{x_{2}-x_{1}}\)

\(\displaystyle m = \frac{5-3}{-2-(-1)} = \frac{2}{-1}\)

\(\displaystyle m = -2\)

Using one set of points or coordinates and the value of slope, plug these values into:

\(\displaystyle y=mx+b\)

Either set of points will give you the same equation of the line.

Coordinates \(\displaystyle (-1,3)\).

\(\displaystyle y-3 = -2(x - (-1)\)

\(\displaystyle y-3 = -2(x+1)\)

Distribute the \(\displaystyle -2\) to what is inside the parenthesis.

\(\displaystyle -2\times x = -2x\)

\(\displaystyle -2 \times 1 = -2\)

\(\displaystyle y-3 = -2x-2\)

Add \(\displaystyle 3\) to both sides of the equation.

\(\displaystyle y-3+3 = -2x-2+3\)

The equation of the line in slope-intercept form  \(\displaystyle y=mx+b\)  is:

\(\displaystyle y = -2x +1\)

The slope-intercept form is \(\displaystyle y = mx+b\), where \(\displaystyle m\) represents the slope, \(\displaystyle x\) and \(\displaystyle y\) represent the points, and \(\displaystyle b\) is the y-intercept or the value of \(\displaystyle y\) when \(\displaystyle x=0.\)

The slope or \(\displaystyle \frac{rise}{run}\) has been given as \(\displaystyle m=-\frac{1}{4}\).

The points that this equation of the line passes through are \(\displaystyle (0,8)\) and

\(\displaystyle b=8\) because based on the points, when \(\displaystyle x=0,y=8.\)  That is the y-intercept.

The equation of the line in slope-intercept form \(\displaystyle y=mx+b\) is:

\(\displaystyle y=-\frac{1}{4} x+8\)

To find the equation of a line given points \(\displaystyle (-5, -3)\) and \(\displaystyle m = -2,\) use the point- slope formula:

\(\displaystyle y-y_{1} = m(x -x_{1})\)

\(\displaystyle y -(-3) = -2 (x--5)\)

\(\displaystyle y +3 = -2(x+5)\)

Distribute the \(\displaystyle -2\) to what is inside the parenthesis.

\(\displaystyle -2(x) = -2x\)

\(\displaystyle -2\times 5 = -10\)

\(\displaystyle y + 3 = -2x-10\)

Subtract \(\displaystyle 3\) from both sides of the equation.

\(\displaystyle y+3-3 = -2x-10-3\)

\(\displaystyle -10-3 = -13\)  When the sign is the same for both integers, add.

The equation of the line in slope-intercept form  \(\displaystyle y=mx+b\)  is:

\(\displaystyle y=-2x-13\)

\(\displaystyle y=mx+b\)

To solve for \(\displaystyle m\) you must use the equation \(\displaystyle (y^{1}-y^{2})/(x^{1}-x^{2})\)

\(\displaystyle (6-(-2))/(1-3) = -8/2 = -4\)

\(\displaystyle y=-4x+b\)

To solve for be we must plug in one of the points

\(\displaystyle 6=-4(1)+b\)

simplify

\(\displaystyle 6=-4+b\)

Add \(\displaystyle 4\) to both sides

\(\displaystyle 10=b\)

\(\displaystyle y=-4x+10\)

To solve this problem you must first find the slope of the original equation by point it into y-mx+b form

\(\displaystyle 4x+y=7\)

subtract x from both sides

\(\displaystyle y=-4x+7\)

Then you must find the reciprocal of the slope to get the slope of the perpendicular line

\(\displaystyle -4\) reciprocal is \(\displaystyle 1/4\)

Finally you must you point-slope form to solve 

\(\displaystyle y-y_{1}=m(x-x_{1})\)

\(\displaystyle y-(3)=1/4(x-(2))\)

Multiply \(\displaystyle 1/4\) through the parentheses

\(\displaystyle y-3=1/4x-1/2\)

add three to both sides

\(\displaystyle y=1/4x+5/2\)