In the equation 3(4x + b) - 2(x - 5) = 2(5x + 8), b is a constant. If the equation has inf… — SAT

In the equation $3(4x + b) - 2(x - 5) = 2(5x + 8)$ , $b$ is a constant. If the equation has infinitely many solutions for $x$ , what is the value of $b$ ?

Answer: 2

Answer

The value of

b

is 2.

Distributing the constants in the equation

3(4x + b) - 2(x - 5) = 2(5x + 8)

yields

12x + 3b - 2x + 10 = 10x + 16

. Combining like terms on the left side simplifies the equation to

10x + 3b + 10 = 10x + 16

. For a linear equation in one variable to have infinitely many solutions, both sides of the equation must be identical. Since the coefficients of

x

are equal (

10 = 10

), the constant terms must also be equal:

3b + 10 = 16

. Solving for

b

yields

3b = 6

, which simplifies to

b = 2

Step-by-Step Solution

Distribute the constants through the parentheses on both sides of the equation

3(4x + b) - 2(x - 5) = 2(5x + 8)

12x + 3b - 2x + 10 = 10x + 16

To eliminate parentheses and allow grouping of like terms.

Combine like terms on the left side of the equation.

10x + 3b + 10 = 10x + 16

To simplify the left-hand expression into the standard linear form.

Equate the constant terms on both sides of the equation.

3b + 10 = 16

A linear equation in one variable has infinitely many solutions when both sides are identical. Since the coefficients of the variable

x

are both 10, the constant terms must be equal.

Solve for

b

by isolating it.

b = 2

Subtracting 10 from both sides gives

3b = 6

, and dividing by 3 yields the final value.

Key Concept

Determining conditions for a linear equation in one variable to have infinitely many solutions (identity).

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