A straight line has the equation ax + by=23. The points (5,-2) and (1,-5) lie on the line. Find the values of a and b

Question

A straight line has the equation ax + by=23. The points (5,-2) and (1,-5) lie on the line. Find the values of a and b

Answer 1

Hi folks, I would solve the problem by first finding the slope of the two points. Then use the point-slope form of the line to find the actual line.

 

Let m be the slope:

 

m = (-5 - (-2)) / (1 - 5)

 

m = 3/4

 

Point-slope form of the line.

 

y - y₁ = m * (x - x₁)

 

m = 3/4

 

y₁ = -5

 

x₁ = 1

 

Substitution:

 

y - (-5) = 3/4 * (x - 1)

 

Multiply the equation by 4 to clear the fraction:

 

4y + 20 = 3x - 3

 

Rearrange:

 

3x - 4y = 23

 

Therefore, a = 3 and b = -4

 

Questions?

Answer 2

Hi Tom.

 

To answer this questions, you will end up with 2 equations and 2 unknowns, we call this a "system of equations".

 

Start with the "generic" expression  ax + by = 23  and use the x- and y-values from each point to create two different equations.  

 

Using (5, -2)   ax + by = 23   becomes 

 

a(5) + b(-2) = 23

 

 

Using (1, -5)    ax + by = 23  becomes

 

a(1) + b(-5) = 23

 

 

Rearranging to look like a typical system of equations we get

 

5a - 2b = 23

1a - 5b = 23

 

Solve the bottom one for "a"

 

Add "5b" to both sides and get

 

a = 23 + 5b

 

Then substitute this "a" into the top equation

 

5a - 2b = 23

 

5(23 + 5b) - 2b = 23

 

Then simplify

 

115 +25b - 2b = 23

 

115 + 23b = 23

 

Subtact 115 from both sides

 

23b = -92

 

Then divide both sides by 23.

 

b = -4

 

Knowing this and also knowing that 

 

a = 23 + 5b

 

substitute b = -4 in for b, and find a

 

 

a = 23 + 5(-4) = 23 - 20 = 3

 

So a = 3

 

To check your answers, plug a=3 and b=(-4) back into the original system of equations and make sure they work in BOTH equations.