First, use the slope formula to find the slope, setting \(\displaystyle x_{1} = -2, y_{1} = 5,x_{2} = -3, y_{2} = 8\).

\(\displaystyle m = \frac{y _{2} - y_{1}}{x _{2} - x_{1}} = \frac{8 -5 }{-3-(-2)} = \frac{3 }{-1} = -3\)

We can write the equation in slope-intercept form as

\(\displaystyle y = mx + b\).

Replace \(\displaystyle m = -3\):

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

We can find \(\displaystyle b\) by substituting for \(\displaystyle x,y\) using either point - we will choose \(\displaystyle (-2, 5)\):

\(\displaystyle 5= -3 (-2) + b\)

\(\displaystyle 5= 6 + b\)

\(\displaystyle 5-6 = 6 -6 + b\)

\(\displaystyle b = -1\)

The equation is \(\displaystyle y = -3x -1\).

First, use the slope formula to find the slope, setting \(\displaystyle x_{1} = 3.5, y_{1} = 5.5,x_{2} = 5.5, y_{2} = -2.5\).

\(\displaystyle m = \frac{y _{2} - y_{1}}{x _{2} - x_{1}} = \frac{-2.5 -5.5}{5.5-3.5} = \frac{-8 }{2} = -4\)

We can write the equation in slope-intercept form as

\(\displaystyle y = mx + b\).

Replace \(\displaystyle m = -4\):

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

We can find \(\displaystyle b\) by substituting for \(\displaystyle x,y\) using either point - we will choose \(\displaystyle (3.5, 5.5)\):

\(\displaystyle 5.5= -4 (3.5) + b\)

\(\displaystyle 5.5= -14 +b\)

\(\displaystyle 5.5+14 = -14 +14+b\)

\(\displaystyle b = 19.5\)

The equation is \(\displaystyle y = -4x + 19.5\).

To find the equation of a line, we need to know the slope and a point that passes through the line.  We can then use the equation \(\displaystyle y-y_1=m(x-x_1)\) where m is the slope of the line, and \(\displaystyle (x_{1}, y_{1})\) a point on the line.  For parallel lines, the slopes are the same.  The slope of \(\displaystyle y=3x+2\) is 3, so the slope of the parallel line will be 3 as well.  We know that the parallel line needs to contain the point (0,1), so we have all of the information we need.  We can now use the equation \(\displaystyle y-y_1=m(x-x_1)\Rightarrow y-1=3(x-0)\Rightarrow y=3x+1.\)

The equation of a line is written in the following format: 

\(\displaystyle y=mx+b\)

1) The first step, then, is to find the slope, \(\displaystyle m\).

\(\displaystyle m\) is equal to the change in \(\displaystyle y\) divided by the change in \(\displaystyle x\).

So,

\(\displaystyle \frac{6-3}{6-4}=\frac{3}{2}\)

\(\displaystyle y=\frac{3}{2}x+b\)

2) Next step is to find \(\displaystyle b\). We can find values for \(\displaystyle x\) and \(\displaystyle y\) from any one of the given points, plug them in, and solve for \(\displaystyle b\).

Let's use (4,3)

So, 

\(\displaystyle 3=\frac{3}{2}(4)+b\)

\(\displaystyle 3=6+b\)

\(\displaystyle b=-3\)

Then we just fill in our value for \(\displaystyle b\), and we have \(\displaystyle y\) as a function of \(\displaystyle x\).

\(\displaystyle y=\frac{3}{2}x-3\)

When an equation is in the \(\displaystyle y=mx+b\) form, its slope is \(\displaystyle m\) and its y-intercept is \(\displaystyle b\). Thus, we need an equation with an \(\displaystyle m\) of 4 and a \(\displaystyle b\) of 6, which would be 

\(\displaystyle y=4x+6\)

Use the point-slope formula to find the equation:

\(\displaystyle Y-Y_{1}=M(X-X_{1})\)

\(\displaystyle y-3=4(x+5)\)

\(\displaystyle y-3 = 4x+20\)

\(\displaystyle y-3+3=4x+20+3\)

\(\displaystyle y=4x+23\)

Start by using point-slope form:

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

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

Multiply the right side by the distributive property:

\(\displaystyle y-5 = -3x -6\) 

Then, convert to standard form:

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

Slope-intercept form is \(\displaystyle y=mx+b\), where \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept.  To rewrite the original equation in slope-intercept form, you must isolate the \(\displaystyle y\) variable: 

\(\displaystyle 5x+20y=60\)

\(\displaystyle -5x+20y=60-5x\)

\(\displaystyle 20y=-5x+60\)

Now, divide each side by 20 so that \(\displaystyle y\) stands alone and simplify, and you are left with the slope-intercept form of the equation:

\(\displaystyle y=-\frac{5}{20}x+\frac{60}{20}\)

\(\displaystyle y=-\frac{1}{4}x+3\)

Let \(\displaystyle P_{1} = (1, \; -2) \; \; and \; \; P_{2} = (-3, \; 6)\).

First, calculate the slope between the two points.

\(\displaystyle m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{6 - (-2)}{-3 - 1} = \frac{8}{-4} = -2\)

Next, use the slope-intercept form to calculate the intercept. We are able to plug in our value for the slope, as well the the values for \(\displaystyle P_1\).

\(\displaystyle y = mx + b\)

\(\displaystyle -2 = -2(1) + b\)

\(\displaystyle b=0\)

Using slope-intercept form, where we know \(\displaystyle m=-2\) and \(\displaystyle b=0\), we can see that the equation for this line is \(\displaystyle y = -2x\).

To find the equation of this line, we need to know its slope and y-intercept. Let's find the slope first using our general slope formula.

\(\displaystyle m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}\)

The points are \(\displaystyle (x_1,y_1)\) and \(\displaystyle (x_2,y_2)\).  In this case, our points are (–3,0) and (2,5). Therefore, we can calculate the slope as the following:

\(\displaystyle m=\frac{5-0}{2-(-3)}=\frac{5}{5}=1\)

Our slope is 1, so plug that into the equation of the line:

\(\displaystyle y=mx+b \,\,\, \rightarrow \,\,\,y=(1)x+b \,\,\, \rightarrow \,\,\, y=x+b\)

We still need to find b, the y-intercept. To find this, we pick one of our points (either (–3,0) or (2,5)) and plug it into our equation. We'll use (–3,0).

\(\displaystyle y=x+b\)

\(\displaystyle 0=-3+b\)

Solve for b.

\(\displaystyle b=3\)

The equation is therefore written as \(\displaystyle y=x+3\).