13. Two linear functions in a coordinate plane have exactly one point of intersection.
Which pair of functions listed below could produce that result?
A, 3x + 4y = 12 and 9x + 12y = 36
B. 3x + 4y = 12 and 9x + 12y = 24
C. 3x + 4y = 12 and x + 4y = 8
D. 3x + 4y = 12 and 6x + 8y = 16

Many solutions if the two functions are Ax + By = C and (m)Ax + (m)By = (m)C, where m, A, B, and c are constant ⇒ both represented by the same line
No solutions if the two functions are Ax + By = C and Ax + By = D, where C ≠ D ⇒ both represented by 2 parallel lines
One solution if the two functions are Ax + By = C and Dx+ Ey = F ⇒ both represented by 2 intersecting lines

Let us check the answers to find which pair of functions could produce two linear functions have exactly one point of intersection

A.

∵ 3x + 4y = 12 ⇒ (1)

∵ 9x + 12y = 36 ⇒ (2)

- Divide (2) by 3

∴ 3x + 4y = 12

The two functions have the same coefficients of x and y and same numerical terms

∴ They are represented by the same line

B.

∵ 3x + 4y = 12 ⇒ (1)

∵ 9x + 12y = 24 ⇒ (2)

- Divide (2) by 3

∴ 3x + 4y = 8

The two functions have the same coefficients of x and y but different in the numerical term

∴ They are represented by two parallel lines

C.

∵ 3x + 4y = 12 ⇒ (1)

∵ x + 4y = 8 ⇒ (2)

The two functions have different coefficients of x and the numerical terms

∴ They are represented by two intersecting lines

D.

∵ 3x + 4y = 12 ⇒ (1)

∵ 6x + 8y = 16 ⇒ (2)

- Divide (2) by 2

∴ 3x + 4y = 8

The two functions have the same coefficients of x and y but different in the numerical terms

∴ They are represented by two parallel lines

The pair of function could produce two linear functions have exactly one point of intersection is 3x + 4y = 12 and x + 4y = 8

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