Question

a(x+b)=4x+10 in the equation above a and b are constants. if the equation has infinitely many solutions for X, what is the value of b?

Answer 1

Answer:

$b= \frac{5}{2}$

Step-by-step explanation:

Given

$a(x+b)=4x+10\\ax+b= 4x+10$

Now we know that system has infinite solution for x

$\frac{a_1}{a_2}=\frac{b_1}{b_2}$

in above equation.

$a_1=a, a_2=4, b_1=ab,b_2=10$

∴$\frac{a}{4}=\frac{ab}{10}\\\therefore b=\frac{10\times a}{4\times a}=\frac{5}{2}$

Answer 2

The value of a and b constants in the given question,according to the given condition is evaluated as:

a = 4,  $b = \dfrac{5}{2}$  

Given that:

• Equation is:  $a(x+b) = 4x+10$

• a and b are constants.
• The equation has infinitely many solutions.

Explanation and evaluation:

The linear equations have infinite solutions when the straight line they're representing are coincident and the y intercept is same.

The equations can be rewritten as:

$a(x+b) = 4x + 10\\ax + ab = 4x + 10$

The intercept is ab on LHS and 10 on RHS. For infinite solutions, we want both intercept same, thus we have ab = 10

The slope of the LHS is a and of RHS is 4, which also needs to be same, thus we have a = 4

Since a = 4 and ab = 10

Thus:

$ab = 10\\\\b = \dfrac{10}{a}\\\\b = \dfrac{10}{4}\\\\b = \dfrac{5}{2}$

Thus,  a = 4,  $b = \dfrac{5}{2}$   is the needed value.

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