Equations of a Line Question 1 Solution

To find the equation of a line passing through the points
( 1, -9 ) and ( -3, -2 ), we must first determine the slope of the line. Using the slope formula, the slope of the line is,

m = ( y₂ - y₁ ) / ( x₂ - x₁ )
m = ( -2 - ( -9 ) ) / ( -3 - 1 )
m = ( -2 + 9 ) / ( -3 - 1 )
m = -7/4

Now, we simply plug in the value of the slope and the x and y values of either point into the point-slope equation.

y - y₁ = m ( x - x₁ )
y - ( -9 ) = ( -7/4 )( x - 1 )
y + 9 = ( -7/4 )( x - 1 )

Hence, the equation of the line is,

y + 9 = ( -7/4 )( x - 1 )

If we had used the second point, then the equation would look like this:

y - y₁ = m ( x - x₁ )
y - ( -2 ) = ( -7/4 )( x - ( -3 ) )
y + 2 = ( -7/4 )( x + 3 )

Thus, the equation of the line would be,

y + 2 = ( -7/4 )( x + 3 )

Although this equation looks different from the previous one, they are, in essence the same equation. To prove this we will convert each equation into slope-intercept form. By doing this, we should come up with the same equation. Converting the first equation would give us,

y + 9 = ( -7/4 )( x - 1 )
y + 9 = ( -7/4 )x + 7/4
y = ( -7/4 )x + 7/4 - 9
y = ( -7/4 )x + 7/4 - 36/4
y = ( -7/4 )x - 29/4

Converting the second equation would give us,

y + 2 = ( -7/4 )( x + 3 )
y + 2 = ( -7/4 )x - 21/4
y = ( -7/4 )x - 21/4 - 2
y = ( -7/4 )x - 21/4 - 8/4
y = ( -7/4 )x - 29/4

As you can see, both equations, when converted into slope-intercept form, are identical. Therefore, any one of the three equations given above would be a correct answer to this problem.