the equation that represents the line that passes through the point (-2, -1) and has a slope of 5.

Question

Not sure what the answer is to this one

Answer 1

Hi -- For this problem, you have already been given some useful information and can use the information to plug in to the standard point-slope formula.  When we talk about point-slope, we mean we have one point defined (-2,-1) and the slope is known (which is 5/1). 

First, we have the point-slope formula:

y-y₁ = m(x-x₁)

Second, we will plug in what we know from what was given in the problem

x₁ = -2  and y₁ = -1

m = 5

Third, let's plug in what we know!

y-(-1) = 5(x-(-2))   Don't forget a (-)(-) = +

y+1 = 5(x+2)         Now, distribute the 5 across (x+2)

y+1 = 5x+10         Now, combine like terms by subtracting 1 from both sides

y = 5x+9              This is the equation of your line with m=5 and y-intercept = (0,9)

Answer 2

Solution: y + 1 = 5 (x+2)  in point slope form   or   y = 5 x + 9  in intercept slope form

PLEASE READ FOLLOWING STEP BY STEP SOLUTION!

STEP 1: Read, understand the situation within, identify and pull out important information.
• There are infinite lines passing through the point (-2,-1), but only one has a slope (m) = 5
• There are several forms for the equation that represents such a line. Let’s consider only two of them: slope intercept form and point slope form.

STEP 2: Translate each of the keywords in the problem to their mathematical symbols.
• Slope Intercept Form equation: y = mx + b; m = slope; b = y-intercept
• Point Slope Form equation: y – y1 = m (x – x1); m = slope; (x1, y1) are the coordinates of the given point (-2,-1).

STEP 3: Set up and solve the equation:

• Point Slope Form equation: y – y1 = m (x – x1); where m = slope; (x1, y1) = (-2,-1)
y – (-1) = 5 (x – (-2)) by substituting the coordinates of the given point
y + 1 = 5 (x+2) ; This is the “point slope form equation” representing the line.

• Slope Intercept Form equation:
y = 5 x + b ; substituting m = 5
-1 = 5 (-2) + b ; substituting the coordinates of the point (-2,-1) to determine b
-1 = -10 + b;
b = 10 – 1 ;
b = 9   y-intercept
y = 5 x + 9 this is the slope intercept form equation that represents the line through (-2,-1) with the slope of 5.

STEP 4: Verify your answer.
• In the Point Slope Form: y + 1 = 5 (x+2)
y + 1 = 5x + 10 by applying the Distributive Property of Multiplication
y1 + 1 = 5x1 + 10 now substitute the coordinates (x1, y1) of the given point (-2, -1)
-1 + 1 = 5*(-2) + 10
0 = -10 +10
0 = 0 This is an identity!, so our point slope equation is a true equation for our line!
• In the Slope Intercept Form equation: y = 5x + 9
y1 = 5x1 + 9 by substituting the coordinates (x1,y1) of the given point (-2,-1)
-1 = 5*(-2) + 9
-1 = -10 + 9
-1 = -1 This is an identity! Our slope intercept equation is a true equation for our line!

STEP 5: Curiosities
• The slope intercept form is a particular case of the point slope form:
y + 1 = 5x + 10 point slope form
y = 5x + 9 by isolating “y”, we get the slope intercept form!
• Graphing the line through (-2, -1) with the slope of 5:
m = RISE / RUN ;
m = 5 ; The slope is positive, so it’s upward going from left to right.
m = 5/1;
RISE = 5; RUN = 1.
Plot the given point (-2,-1) on a Cartesian plane. Starting on this point go up 5 units (RISE =5), and then to the right 1 unit (RUN = 1). You get the point (-1,4).
Remember: RISE = change of “y”
RISE = y2 – y1 ; so,
y2 = RISE + y1
y2 = 5 + (-1) ;
y2 = 4   this is the y-coordinate of the second point!
Do the same for the RUN and you get x2 = RUN – x1
x2 = 1 + (-2)
x2 = -1 this is the x-coordinate of the second point.
So the second point is P2 = (-1,4).
Finally: draw a line connecting the two points (-2,-1) and (-1,4). There you go!

Answer 3

The point-slope formula for a line through (x₀,y₀) and having slope m is:

y = m(x - x₀) + y₀

You can plug in the information x₀ = -2, y₀ = -1, m = 5 and simplify.

y = 5(x + 2) - 1

y = 5x + 10 - 1

y = 5x + 9

Answer 4

For this you can use the classic slope-intercept formula:

y = mx + b

They give you a point with x and y coordinates and they give you the slope so you just have to solve for b (also known as the y-intercept):

-1 = (5*-2) + b

-1 = -10 + b

9 = b

Now you just use the values of m = 5 and b = 9 to rewrite the equation as:

 y = 5x + 9