If we chose to use the slope formula to determine a possible slope, we have:

\(\displaystyle m= \frac{y_2-y_1}{x_2-x_1} = \frac{6-3}{1-1} = \frac{3}{0}\)

Our slope becomes undefined.  The x-values will not change, and we are not able to write this in the slope-intercept form.

This line is a vertical line, and will be represented by \(\displaystyle x=1\) because the x-values will not change in a vertical line.

The answer is: \(\displaystyle x=1\)

To determine the equation for the line given only a point on the line and its slope, we can use point-slope form, which is given by the following:

\(\displaystyle y-y_{1}=m(x-x_{1})\), where \(\displaystyle m\) is the slope of the line and \(\displaystyle (x_{1}, y_{1})\) is the point on the line.

Using the formula, we get

\(\displaystyle y-5=4(x-1)\), which simplified becomes \(\displaystyle y=4x+1\).

Write the slope-intercept form for linear equations.

\(\displaystyle y=mx+b\)

The \(\displaystyle m\) is the slope of the equation, and \(\displaystyle b\) is the y-intercept.

Substitute the values into the equation.

The equation of the line is:  \(\displaystyle y=3x+3\)

Write the formula for point-slope form.

\(\displaystyle y-y_1 = m(x-x_1)\)

The variable \(\displaystyle m\) is the slope, and the point is in \(\displaystyle (x_1,y_1)\) format.  

\(\displaystyle m=5\)

\(\displaystyle (x_1,y_1)=(5,3)\)

Substitute all the givens.   There's is no need to simplify.

The answer is:  \(\displaystyle y-3 = 5(x-5)\)

Write the slope intercept form.

\(\displaystyle y=mx+b\)

Substitute the slope and the point to find the y-intercept, \(\displaystyle b\).

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

Solve for the unknown variable.

\(\displaystyle 3=9+b\)

Subtract nine from both sides.

\(\displaystyle -6=b\)

Write the equation now that we have the slope and y-intercept.

The answer is: \(\displaystyle y=3x-6\)

Write the slope intercept form.  The equation will be in this form.

\(\displaystyle y=mx+b\)

Write the slope formula.

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

Let \(\displaystyle (x_1,y_1)=(1,9)\) and \(\displaystyle (x_2,y_2)=(-2,11)\).  

Substitute the points into the slope formula.

\(\displaystyle m = \frac{y_2-y_1}{x_2-x_1} = \frac{11-9}{-2-1} = \frac{2}{-3}\)

The slope is:  \(\displaystyle -\frac{2}{3}\)

Use the slope and a given point, substitute them into the slope intercept form to find the y-intercept.

\(\displaystyle 9=(-\frac{2}{3})(1)+b\)

Solve for the y-intercept.

\(\displaystyle 9=-\frac{2}{3}+b\)

Add two-thirds on both sides.

\(\displaystyle 9+ \frac{2}{3}=-\frac{2}{3}+b+ \frac{2}{3}\)

Simplify both sides.

\(\displaystyle b=9\frac{2}{3}\)

With the slope and y-intercept known, write the formula.

The answer is:  \(\displaystyle y=-\frac{2}{3}x+9\frac{2}{3}\)

The x-intercept is the value when \(\displaystyle y=0\).

The y-intercept is the value when \(\displaystyle x=0\).

We know the two points needed to find the equation of the line.

The points are:  \(\displaystyle (7,0)\) and \(\displaystyle (0,4)\)

Use the slope formula to find the slope.

\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1} = \frac{4-0}{0-7} = -\frac{4}{7}\)

Write the slope-intercept equation.

\(\displaystyle y=mx+b\)

Substitute the slope and the y-intercept.

The answer is:  \(\displaystyle y=-\frac{4}{7}x+4\)

Write the equation in point-slope form.

\(\displaystyle y-y_1=m(x-x_1)\)

The variable \(\displaystyle m\) is the slope.  Substitute the slope and the point.

\(\displaystyle y-2 = 4(x-(-5))\)

Simplify the right side of the equation.

The answer is:  \(\displaystyle y-2 = 4(x+5)\)

The equation for a line is always \(\displaystyle \small y = mx + b\), where \(\displaystyle \small m = slope\) and \(\displaystyle \small b = y-intercept\).

Given two data points, we are able to find the slope, m, using the formula 

\(\displaystyle \small \frac{y_{1}-y_{2}}{x_{1}-x_{2}}\).

Using the data points provided, our formula will be: 

\(\displaystyle \small \small \frac{3-9}{5-3}\), which gives us \(\displaystyle \small \small -\frac{6}{2} ,\) or , \(\displaystyle \small \small m = -3\).

Our y-intercept is given.

Thus our equation for the line containing these points and that y-intercept is \(\displaystyle \small \small y = -{3}x + 3\).