Question

Solve the equation \(2(x+3)-1=-7\).

Short answer

x = -6.

Step-by-step solution

- Simplify the Equation

First, distribute the 2 on the left side of the equation: \[2(x + 3) - 1 = -7\] becomes \[2x + 6 - 1 = -7\].

- Combine Like Terms

Next, combine the constants on the left side. \[2x + 6 - 1\] simplifies to \[2x + 5\], resulting in \[2x + 5 = -7\].

- Isolate the Variable Term

Subtract 5 from both sides to isolate the variable term: \[2x + 5 - 5 = -7 - 5\] which simplifies to \[2x = -12\].

- Solve for x

Finally, divide both sides by 2 to solve for x: \[\frac{2x}{2} = \frac{-12}{2}\] which simplifies to \[x = -6\].

Key concepts

Distributive Property

The distributive property is a foundational concept in algebra. It allows us to simplify expressions by distributing a number over terms in parentheses. For example, in the equation \[2(x + 3) - 1 = -7\], we apply the distributive property to \[2(x + 3)\] to get \[2x + 6\]. This property is expressed as \[a(b + c) = ab + ac\]. By distributing, we make the equation easier to work with. Always ensure you distribute the number to each term inside the parentheses.

Combining Like Terms

Combining like terms means summing up terms with the same variable and constant terms. After applying the distributive property, we had \[2x + 6 - 1 = -7\]. Here, \[6\] and \[-1\] are constants and can be combined. The equation simplifies to \[2x + 5 = -7\]. Always identify and combine like terms to make the equation cleaner and simpler to solve.

Isolating the Variable

Isolating the variable involves manipulating the equation to get the variable term alone on one side. From \[2x + 5 = -7\], we subtract \[5\] from both sides to isolate \[2x\], giving us \[2x = -12\]. This step is crucial for solving the equation, as it prepares us for finding the value of the variable. Always perform the same operation on both sides to maintain equality.

Solving for x

Finally, solving for \[x\] means finding the value that makes the equation true. After isolating the variable, we need to get \[x\] by itself. From \[2x = -12\], we divide both sides by \[2\], resulting in \[x = -6\]. At this stage, we've simplified the equation to its final form, solving for \[x\]. Always check your work by substituting the solution back into the original equation to verify it satisfies the equation.