The easiest way to determine the slope and y-intercept of a line is by rearranging its equation to the $\displaystyle y=mx+b$ form. In this form, the slope is the $\displaystyle m$ and the y-intercept is the $\displaystyle b$.

Rearranging $\displaystyle 2y-4x=12$

gives you

$\displaystyle y=2x+6$

which has an $\displaystyle m$ of 2 and a $\displaystyle b$ of 6.

When finding the equation of a line given two or more points, the first step is to find the slope of that line. We can use the slope equation, $\displaystyle \frac{y_{2}-y_{_{1}}}{x_{2}-x_{1}}$. Any combination of the three points can be used, but let's consider the first two points, $\displaystyle (1,6)$ and $\displaystyle (11,12)$.

 $\displaystyle \frac{12-6}{11-1} = \frac{3}{5}$

So $\displaystyle \frac{3}{5}$ is our slope.

Now, we have the half-finished equation

 $\displaystyle y = \frac{3}{5}x + b$

and we can complete it by plugging in the $\displaystyle x$ and $\displaystyle y$ values of any point. Let's use $\displaystyle (1,6)$.

Solving

 $\displaystyle 6 = \frac{3}{5}(1)+b$ 

for $\displaystyle b$ gives us

 $\displaystyle b = 6 - \frac{3}{5}$

so

 $\displaystyle b = 5\frac{2}{5} = \frac{27}{5}$

We now have our completed equation: 

$\displaystyle y = \frac{3}{5}x + \frac{27}{5}$

We need to find the equation of the line in slope-intercept form.

$\displaystyle y=mx+b$

In this formula, $\displaystyle m$ is equal to the slope and $\displaystyle b$ is equal to the y-intercept.

To find this equation, first, we need to find the slope by using the formula for the slope between two point.

$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$

In the formula, the points are $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$.  In our case, the points are $\displaystyle (-3,-1)$ and $\displaystyle (4,9)$. Using our values allows us to solve for the slope.

$\displaystyle m=\frac{9-(-1)}{4-(-3)}=\frac{10}{7}$

We can replace the variable $\displaystyle m$ with our new slope.

$\displaystyle y=\frac{10}{7}x+b$

Next, we need to find the y-intercept. To find this intercept, we can pick one of our given points and use it in the formula.

$\displaystyle (-3,-1)$

$\displaystyle -1=\frac{10}{7}(-3)+b$

Solve for $\displaystyle b$.

$\displaystyle -1=\frac{-30}{7}+b$

$\displaystyle b=-1+\frac{30}{7}$

$\displaystyle b=\frac{23}{7}$

Now, the final equation connecting the two points can be written using the new value for the y-intercept.

$\displaystyle y=\frac{10}{7}x+\frac{23}{7}$

Since all of the answers are in the $\displaystyle y=mx+b$ form, the slope of each line is indicated by its $\displaystyle m$ and its y-intercept is indicated by its $\displaystyle b$. Thus, a line with a slope of $\displaystyle -3$ and a y-intercept of $\displaystyle 4$ must have an equation of $\displaystyle y=-3x+4$.

The expression under the radical must be $\displaystyle \geq 0$.  Hence

$\displaystyle \sqrt{2x-5} \geq 0$

Solving for $\displaystyle x$, we get

$\displaystyle x\geq \frac{5}{2}$

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 

$\displaystyle y=4x+6$