How do you write an equation in standard form for a line passing through (5,22) and (3,12)?

2 Answers
Mar 3, 2018

Use the points to write the equation in slope-intercept form.
Then write that equation in standard form.

Standard form for these points is

color(white)(.....)5x - y = 3

Explanation:

To write an equation in standard form when you have two points, first write the equation in slope-intercept form.

"Write the equation in slope-intercept form"
color(white)(........)y = mx + b

color(white)(.........)―――――――

First find m, the slope

Slope = ((y - y'))/((x - x'))

1) Decide which point will be (x,y)

Assign (5,22) to be (x,y)
Assign (3,12) to be (x',y')

You can choose either point to be (x',y') and it will come out the same, but if you select wisely, you can sometimes avoid working with negative numbers.

2) Sub in the values for the variables

((y - y'))/((x - x'))

((22 - 12))/((5 - 3))

3) Do the subtractions to find the slope m

color(white)(.....)(10)/(2) =5 larr value for slope m

4) So now the equation so far is

color(white)(.....)y = 5x + b

color(white)(.........)―――――――

Next find b, the y intercept

1) Using either given point, sub in the values for (x,y) and solve for b

color(white)(.......) y = 5  x  + b
color(white)(.....)12 = 5(3) + b

2) Clear the parentheses
12 = 15 + b

3) Subtract 15 from both sides to isolate b
-3 = b

So the equation in point-slope form is
color(white)(.....)y = 5x - 3

color(white)(.........)―――――――

Now put the equation in standard form

color(white)(.......)ax + by = c
where a is a positive integer

color(white)(.....)y = 5x - 3

1) Subtract 5x from both sides to isolate the constant

-5x + y = - 3

2) Multiply through by -1 to clear the minus sign on x

5x - y = 3 larr standard form

Mar 3, 2018

5x-y=3 same as y=5x-3

Explanation:

color(blue)("Preamble")

Gradient (slope) is:

m=(y-("known value for "y))/(x-("known value of "x)) ..Equation(1)

Method
Step 1: determine m from the given points.
Step 2: use one of the given points to substitute values for x and y into Equation(1)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Answering the question")

color(brown)("Step 1:")

Always read left to right on the x-axis for this:

Set the left most point as P_1->(x_1,y_1)=(3,12)
Set the rightmost point as P_2->(x_2,y_2)=(5,22)

Set the gradient (slope) as m

m=("change in the y-axis")/("change in the x-axis") ->(y_2-y_1)/(x_2-x_1)=(22-12)/(5-3)=10/2

But 10/2 is the same as 5/1->5

m=5/1 larr" Deliberately written this way"
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(brown)("Step 2:")

Using P_1

m=("change in the y-axis")/("change in the x-axis")->m=5/1=(y-12)/(x-3)

Cross multiply giving:

5(x-3)=1(y-12)

5x-15=y-12

5x-y=15-12

5x-y=3

Tony BTony B