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Math  /  Algebra

Questioni-Ready JAYDEN
Which statement describes the solution to the equation a+5(2a−1)+3=11a−2a+5(2 a-1)+3=11 a-2 ? The equation has no solution. The equation has exactly one solution, a=211a=\frac{2}{11}. The equation has exactly one solution, a=−4a=-4. The equation has infinitely many solutions.

Studdy Solution

STEP 1

1. We are given the equation a+5(2a−1)+3=11a−2 a + 5(2a - 1) + 3 = 11a - 2 .
2. We need to determine the nature of the solution(s) to this equation.

STEP 2

1. Simplify the equation.
2. Solve for a a .
3. Determine the nature of the solution.

STEP 3

Expand the expression 5(2a−1) 5(2a - 1) in the equation.
5(2a−1)=10a−5 5(2a - 1) = 10a - 5
Substitute back into the equation:
a+10a−5+3=11a−2 a + 10a - 5 + 3 = 11a - 2

STEP 4

Combine like terms on the left side of the equation.
(a+10a)+(−5+3)=11a−2 (a + 10a) + (-5 + 3) = 11a - 2
11a−2=11a−2 11a - 2 = 11a - 2

STEP 5

Subtract 11a 11a from both sides of the equation.
11a−2−11a=11a−2−11a 11a - 2 - 11a = 11a - 2 - 11a
−2=−2 -2 = -2

STEP 6

The equation simplifies to a true statement −2=−2-2 = -2, which means the equation holds for all values of a a .
Therefore, the equation has infinitely many solutions.
The correct statement is:
The equation has infinitely many solutions.

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