Examkin

Question

Difficulty: Very hardLinear Equations in One Variable
In the equation below, aa and bb are constants.
a(3x−5)−b(2x+2)=4x−12a(3x - 5) - b(2x + 2) = 4x - 12
If the equation has infinitely many solutions for xx, what is the value of a+ba + b?
  1. 3Answer
  2. B
    7
  3. C
    8
  4. D
    13

Answer

3
To find the value of a+ba + b that results in infinitely many solutions, we first distribute the constants aa and bb to rewrite the equation as (3a−2b)x−(5a+2b)=4x−12(3a - 2b)x - (5a + 2b) = 4x - 12. For a linear equation to have infinitely many solutions, the coefficient of xx on both sides must be equal, and the constant terms on both sides must be equal. This gives the system of equations 3a−2b=43a - 2b = 4 and 5a+2b=125a + 2b = 12. Adding these two equations yields 8a=168a = 16, which gives a=2a = 2. Substituting a=2a = 2 into 3a−2b=43a - 2b = 4 yields 6−2b=46 - 2b = 4, which gives b=1b = 1. Therefore, the value of a+ba + b is 2+1=32 + 1 = 3.

Step-by-Step Solution

1
Distribute the constants aa and bb on the left side of the equation and group the terms.
(3a−2b)x−(5a+2b)=4x−12(3a - 2b)x - (5a + 2b) = 4x - 12
To write the linear equation in the standard form Ax+B=Cx+DAx + B = Cx + D so that coefficients can be compared.
2
Set up a system of equations by equating the coefficient of xx and the constant term on both sides of the equation.
3a−2b=43a - 2b = 4 and −(5a+2b)=−12-(5a + 2b) = -12 (which simplifies to 5a+2b=125a + 2b = 12)
For a linear equation in one variable to have infinitely many solutions, the coefficient of xx on both sides must be equal, and the constant terms on both sides must be equal.
3
Solve the system of equations by adding them to eliminate bb.
8a=16  ⟹  a=28a = 16 \implies a = 2
Adding 3a−2b=43a - 2b = 4 and 5a+2b=125a + 2b = 12 eliminates bb, allowing us to solve directly for aa.
4
Substitute a=2a = 2 back into 3a−2b=43a - 2b = 4 to solve for bb.
3(2)−2b=4  ⟹  6−2b=4  ⟹  −2b=−2  ⟹  b=13(2) - 2b = 4 \implies 6 - 2b = 4 \implies -2b = -2 \implies b = 1
Substituting the value of aa allows us to determine bb.
5
Calculate the value of a+ba + b.
a+b=2+1=3a + b = 2 + 1 = 3
To find the sum of the two constants as requested.

Key Concept

Linear Equations in One Variable (Infinitely Many Solutions)
Rate this question

Topics

Linear Equations in One Variable

More questions from this topic