# ``passes through (0, -2) and (4, 2)`` Write an equation in slope-intercept form that satisfies each set of conditions.

Slope of a line passing through two points (a,b) & (c,d) is given by the equation,

m = (d-b)/(c-a)

& equation of the line is :-

y - b = m(x-a)

The standard for of representing a line in slope-intercept form is :-

y = mx + b ; where m = slope of the line & 'b' = y-intercept

Now, the line passes through the points (0,-2) & (4,2)

Thus, slope of the line = (2-(-2))/(4-0) = 4/4 = 1

Now, equation of the line is :-

y - (-2) = 1*(x-0)

or, y + 2 = x

or, y = x - 2 = slope-intercept representation of the line

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To find the slope-intercept equation for a line passing through the points (0,-2) and (4,2) we must first find the slope of the line. We will say that point (0,-2)=(x₁,y₁)  and the point (4,2)=(x₂-y₂).

We can then use the following equation to find our slope:

m(slope)= (y₂-y₁)/(x₂-x₁)

m=2-(-2)/4-0

m=4/4, therefore m=1.

Now we can choose to use either of the two points and plug it into the slope-intercept equation y=mx+b(y-intercept).

I am going to use the point (4,2) to solve our equation, but if you choose the point (0,-2), you will reach the same answer.

2=(1)(4)+b

2=4+b

-4  -4

———

-2=b

Now that we know the y-intercept (where the line crosses the y axis), we can go back and put what we know into writing the slope-intercept equation. We will leave the variables x and y in the equation so that we can later find other points on the same line if we need to.  Therefore, our equation will be : y= 1x + (-2) or y=x-2.

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` `  To write an equation in slope intercept form (y=mx + b) for the line that passes through  the points (0, -2) and (4, 2), we must find determine the slope and y-intercept of the line.

Slope (m) is defined as the steepness of the line and can be written as the ratio of the line's rise over its run.   The formula is `(y_(2)-y_(1))/(x_(2)-x_(1)).`` `

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` `By substituting the x and y values from the ordered pairs into the formula and simplifying as shown, we can determine the slope if the line passing through these points is 1.   Therefore, m = 1.

The y-intercept (b) is the point where the line crosses the y-axis on the coordinate plane.   This point will always be (0, y).   Since one of the points given to us is in this form, we already know the y-intercept is -2.

To verify this, use the point-slope formula.

Choose one of the points to substitute in for the  `y_(1)` and `x_(1)`  and use the slope you found in the previous step to substitute in for `m`.  Then simplify and solve for y.

`(y-2) = 1(x-4)`

`(y-2) = x -4 `

`y = x - 2`

This is the equation of the line in slope intercept form.   We know the slope of the line is 1 and this equation confirms for us that the y-intercept is -2.

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