Question

Ax +by = 16

3x+2y=64

In the system of equations, a and b are constants. if the system has infinitely many solutions, what's the value of ab?

Answer 1

• The system has infinitely many solutions when a = 3/4 and b = 1/2

Given

System of linear equations

• ax +by = 16
• 3x+2y=64

To find

• Values of a and b that leads to infinitely many solutions

Solution

The linear system of equations has infinitely many solutions when lines overlap, so both have same slope and y-intercept.

We can solve it in two different ways

1. Note that when the first equation is multiplied by 4, it has same value of constant as the second equation, which is 64.

• 4ax + 4by = 64
• 3x + 2y = 64

When compared it gives us:

• 4a = 3 ⇒ a = 3/4
• 4b = 2 ⇒ b = 2/4 = 1/2

2. Convert both equations from standard to slope-intercept form and compare:

• ax + by = 16
• by = -ax + 16
• y = - (a/b)x + 16/b

• 3x + 2y = 64
• 2y = - 3x + 64
• y = - (3/2)x + 32

Work out values of a and b:

• 16/b = 32 ⇒ b = 16/32 ⇒ b = 1/2
• - (a/b) = - 3/2 ⇒ a/b = 3/2 ⇒ a = 3b/2 ⇒ a = 3(1/2)/2 ⇒ a = 3/4