Solve each equation for x. 2(x-a) +b =3x+a

Verified step by step guidance

1. Start by expanding the left side of the equation: distribute the 2 to both terms inside the parentheses, resulting in $2x-2a+b=3x+a$.
2. Next, move all terms involving $x$ to one side of the equation and constant terms to the other side. Subtract $2x$ from both sides to get $-2a+b=x+a$.
3. Subtract $a$ from both sides to isolate terms involving $x$ on one side: $-2a+b-a=x$.
4. Combine like terms on the left side: $-3a+b=x$.
5. The equation is now solved for $x$, which is expressed as $x=-3a+b$.

Key Concepts

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property allows us to multiply a single term by two or more terms inside a set of parentheses. In the context of the given equation, applying the distributive property helps to simplify expressions like 2(x - a) into 2x - 2a, making it easier to isolate the variable x.

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. In the equation 2(x - a) + b = 3x + a, it is essential to combine terms on both sides to isolate x. This process helps in reducing the equation to a simpler form, facilitating the solution.

Isolating the Variable

Isolating the variable means rearranging an equation to get the variable (in this case, x) on one side by itself. This often involves performing inverse operations, such as addition, subtraction, multiplication, or division. In the equation provided, after simplifying and combining like terms, the goal is to manipulate the equation to express x in terms of other constants and variables.