At first glance, it may seem that there is not enough information to determine the equation for either line. Notice, however, that the blue line passes through the origin, so we therefore know two points which lie on the blue line. These points are (0,0) and (2,4). Let's use these points to determine the slope of the blue line as such: \(\displaystyle m_{blue}=\frac{4-0}{2-0}=2\).

Now, we may use this slope, to determine the slope of the perpendicular, red line. The slope of the red line is the negative reciprocal of the slope of the blue line because they are normal to one another. Therefore, the slope of the red line is 

\(\displaystyle m_{red}=-\frac{1}{2}\).

Additionally, we are given a point through which the red line passes. This point is (0,3). We may immediately identify that this point represents the y-intercept, and so the y-intercept of the red line is \(\displaystyle b_{red}=3\). 

Let's now use the information about the red line to write the equation of that line in slope-intercept form as: 

\(\displaystyle y=-\frac{1}{2}x+3\).

When finding the equation of a line given two points, we must first find the slope of the line.  To find the slope of a line in slope-intercept form

\(\displaystyle y = mx + b\)

where m is the slope, we calculate

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

where \(\displaystyle (x_1, y_1)\) and \(\displaystyle (x_2, y_2)\) are the points.  Given the points, we can substitue. We get

\(\displaystyle m = \frac{-1 - 5}{-2 - 4}\)

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

\(\displaystyle m = 1\)

Now that we know the slope, we can substitute the slope into the slope-intercept form

\(\displaystyle y = 1x + b\)

All we need to do now if find the value of b.  To do that, we will substitute one of the points into the equation.  Let's use (4, 5).  So,

\(\displaystyle 5 = 1 \cdot 4 + b\)

\(\displaystyle 5 = 4 + b\)

\(\displaystyle 1 = b\)

\(\displaystyle b = 1\)

So if we substitute the value of b into the slope-intercept form with the slope included, we get

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

which can also be written as 

\(\displaystyle y = x + 1\)

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:  \(\displaystyle y-2 = 4(x+5)\)