The slope intercept form states that \(\displaystyle \dpi{100} \small y=mx+b\). In order to convert the equation to the slope intercept form, isolate \(\displaystyle \dpi{100} \small y\) on the left side:

\(\displaystyle \dpi{100} \small 8x-2y=12\)

\(\displaystyle \dpi{100} \small -2y=-8x+12\)

\(\displaystyle \dpi{100} \small y=4x-6\)

The equation of a line is

y=mx + b where m is the slope

Rearrange the equation to match this:

7x + 28y = 84

28y = -7x + 84

y = -(7/28)x + 84/28

y = -(1/4)x + 3

m = -1/4

First solve for the slope of the line, m using y=mx+b

m = (y2 – y1) / (x2 – x1)

= (15 – 14) / (–5 –3)

= (1 )/( –8)

=–1/8

y = –(1/8)x + b

Now, choose one of the coordinates and solve for b:

14 = –(1/8)3 + b

14 = –3/8 + b

b = 14 + (3/8)

b = 14.375

y = –(1/8)x + 14.375

Our first step will be to determing the slope of the line that connects the given points.

\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-5}{2-3}=\frac{1}{-1}=-1\)

Our slope will be \(\displaystyle -1\). Using slope-intercept form, our equation will be \(\displaystyle y=(-1)x+b\). Use one of the give points in this equation to solve for the y-intercept. We will use \(\displaystyle \dpi{100} \small (2,6)\).

\(\displaystyle 6=(-1)(2)+b\)

\(\displaystyle 6=-2+b\)

\(\displaystyle 8=b\)

Now that we know the y-intercept, we can plug it back into the slope-intercept formula with the slope that we found earlier.

\(\displaystyle y=(-1)x+8\)

\(\displaystyle y=-x+8\)

This is our final answer.

The answer is \(\displaystyle x^2+y=10\).

A line can only be represented in the form \(\displaystyle x=z\) or \(\displaystyle y=mx+b\), for appropriate constants \(\displaystyle z\), \(\displaystyle m\), and \(\displaystyle b\). A graph must have an equation that can be put into one of these forms to be a line.

\(\displaystyle x^2+y=10\) represents a parabola, not a line. Lines will never contain an \(\displaystyle x^2\) term.

If we rearrange the second equation it is the same as the first equation. They are the same line.

For finding the equation of a line, we will be using point-slope form, which is

\(\displaystyle y-y_0=m(x-x_0)\), where \(\displaystyle m\) is the slope, and \(\displaystyle (x_0, y_0)\) is a point. 

\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-3}{-10-0}=\frac{1}{-10}=-\frac{1}{10}\)

We will pick the point \(\displaystyle (0,3)\)

\(\displaystyle y-3=-\frac{1}{10}(x-0)\)

\(\displaystyle y=-\frac{1}{10}x+3\)

If we picked the point \(\displaystyle (-10,4)\)

\(\displaystyle y-4=-\frac{1}{10}(x+10)\)

\(\displaystyle y=-\frac{1}{10}x-1+4\)

\(\displaystyle y=-\frac{1}{10}x+3\)

We get the same result

Since we want a line that is parallel, we will have the same slope as the line \(\displaystyle (-3)\). We can use point slope form to create an equation.

\(\displaystyle y-y_0=m(x-x_0)\), where \(\displaystyle m\) is the slope and \(\displaystyle (x_0, y_0)\) is a point.

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

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

\(\displaystyle y-5=-3x-6\)

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

Based on the graph the y-intercept is 4. So we can eliminate choice y = x/2 - 4.

The graph is rising to the right which means our slope is positive, so we can eliminate choice y = -1/2x + 4.

Based on the line, if we start at (0,4) and go up 1 then 2 to the right we will be back on the line, meaning we have a slope of  (1/2).

Using the slope intercept formula we can plug in y= (1/2)x + 4.

Let P₁ = (1, 4) and P₂ = (5, 1)

Substitute these values into the distance formula: 

The distance formula is an application of the Pythagorean Theorem:  a² + b² = c²