To solve this question, you could use two points such as (1.2,0) and (0,-4) to calculate the slope which is 10/3 and then read the y-intercept off the graph, which is -4.

P₁ (0, 6) and P₂ (4, 0)

First, calculate the slope:  m = rise ÷ run = (y₂ – y₁)/(x₂ – x₁), so m = –3/2

Second, plug the slope and one point into the slope-intercept formula: 

y = mx + b, so 0 = –3/2(4) + b and b = 6

Thus, y = –3/2x + 6

If P₁(1, 3) and P₂(3, 6), then calculate the slope by m = rise/run = (y₂ – y₁)/(x₂ – x₁) = 3/2

Use the slope and one point to calculate the intercept using y = mx + b

Then convert the slope-intercept form into standard form.

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.

In finding the equation of the line given its slope and a point through which it passes, we can use the slope-intercept form of the equation of a line:

\(\displaystyle y=mx+b\), where \(\displaystyle m\) is the slope of the line and \(\displaystyle b\) is its \(\displaystyle y\)-intercept.

Plug the given conditions into the equation to find the \(\displaystyle y\)-intercept.

\(\displaystyle 1=2(8)+b\)

Multiply:

\(\displaystyle b+16=1\)

Subtract \(\displaystyle 16\) from each side of the equation:

\(\displaystyle b=-15\)

Now that you have solved for \(\displaystyle b\), you can write out the full equation of the line:

\(\displaystyle y=2x-15\)