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Math  /  Algebra

QuestionSolve the equation. 6−2(a+1)=7+aa=□\begin{array}{l} 6-2(a+1)=7+a \\ a=\square \end{array}

Studdy Solution

STEP 1

1. The equation 6−2(a+1)=7+a 6 - 2(a+1) = 7 + a is a linear equation in one variable, a a .
2. The goal is to isolate the variable a a on one side of the equation.
3. The equation involves basic algebraic operations, including distribution and combining like terms.

STEP 2

1. Simplify the equation by applying the distributive property.
2. Combine like terms on each side of the equation.
3. Isolate the variable a a by performing algebraic operations.
4. Verify the solution by substituting it back into the original equation.

STEP 3

Apply the distributive property to the term −2(a+1) -2(a+1) :
6−2a−2=7+a 6 - 2a - 2 = 7 + a

STEP 4

Combine like terms on the left side of the equation:
(6−2)−2a=7+a (6 - 2) - 2a = 7 + a
This simplifies to:
4−2a=7+a 4 - 2a = 7 + a

STEP 5

To isolate a a , first move all terms involving a a to one side and constant terms to the other side. Add 2a 2a to both sides:
4=7+a+2a 4 = 7 + a + 2a
This simplifies to:
4=7+3a 4 = 7 + 3a

STEP 6

Subtract 7 from both sides to isolate terms with a a :
4−7=3a 4 - 7 = 3a
This simplifies to:
−3=3a -3 = 3a

STEP 7

Divide both sides by 3 to solve for a a :
−33=a \frac{-3}{3} = a
This simplifies to:
a=−1 a = -1

STEP 8

Verify the solution by substituting a=−1 a = -1 back into the original equation:
Original equation: 6−2(a+1)=7+a 6 - 2(a+1) = 7 + a
Substitute a=−1 a = -1 :
6−2(−1+1)=7+(−1) 6 - 2(-1+1) = 7 + (-1)
Simplify both sides:
6−2(0)=7−1 6 - 2(0) = 7 - 1
6=6 6 = 6
Both sides are equal, confirming that the solution is correct.
The solution is:
a=−1 a = \boxed{-1}

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