What is the equation

Question

What is the equation of the line that passes through the points (-2,-3) and (4,-5)?

Answer 1

The equation in which they refer to in the question is the slope-intercept equation y=mx+b where m is the slope and b is the y-intercept. So, to answer this question given two points we would need to solve the slope-intercept equation. To do this we need to

1) Solve for the slope m,

using the point-slope formula (y2-y1)/(x2-x1), where (x1,y1) & (x2,y2) are the two

points given in the question

2) Solve for b the y-intercept,

using the slope-intercept formula: y-y1 = m(x - x1)

To solve for b in the slope-intercept equation y=mx+b after we solved for m (the slope) in step 1 we use the slope-intercept formula. (do not confuse the slope-intercept equation or slope-intercept form for with slope-intercept formula. The formula is just a recipe or calculation to follow to get the values needed to create the slope-intercept equation or slope-intercept form)

1) To solve for the slope given any two points we would use the following point slope formula

(y2 - y1) /( x2 - x1)

for our question

x1 = -2

y1 = -3

x2 = 4

y2 = -5, so

(-5 - -3) / (4 - -2) = (-5 + 3) /( 4 + 2) = -2/6 = -1/3

So, the slope m is -1/3

2) To solve for b the y-intercept using a point we would use the following slope-intercept formula

y-y1 = m(x - x1)

To solve the slope-intercept formula we would need

-m (the slope we just calculated as -1/3)

-x1 and y1 (The point (-2,-3) that was given in the problem question)

plugging m, x1 and y1 into the formula we get

y - -3 = -1/3(x - -2) = y + 3 = -1/3(x+2) = y + 3 = -1/3x - 2/3

So we are solving the slope-intercept form

y=mx+b

using y + 3 = -1/3x - 2/3 we need to factor out the +3 on the left side of the equation to get y alone on the left side. To do that we subtract 3 from each side to get

y + 3 -3 = -1/3x - 2/3 - 3

y + (3 - 3) = -1/3x + (-2/3 - 3) Note: -3 = -9/3

y + 0 = -1/3x + (-11/3) Note: -2/3 + -9/3 = -11/3

y = -1/3x - 11/3 Note: we don't need the + sign after -1/3x as adding a single negative number is -

so the final answer is

y = -1/3x - 11/3, where m = -1/3 and the y-intercept b is -11/3

In Conclusion:

By solving the slope m using point-slope formula and then solving the slope-intercept formula using one set of points you can solve the equation or more precisely the slope-intercept equation.

To double check that our answer is correct

plug in the second point that was given (4,-5) and see if the equation is true

y = -1/3x - 11/3, so x =4 and y needs to equal -5

-5 = -1/3(4) - 11/3

-5 = -4/3 - 11/3

-5 = -15/3

-5 = -5

So, the equation is correct

Answer 2

Start with two-point form:

(y + 3) / (x + 2) = (-3 + 5) / (-2 - 4)