Question

Solve each equation. \(6 x-2(x-4)=24\)

Short answer

x = 4

Step-by-step solution

- Distribute the terms inside the parenthesis

First, distribute the -2 to both terms inside the parenthesis:\[6x - 2(x - 4) = 24\] This gives us \[6x - 2x + 8 = 24\]

- Combine like terms

Next, combine like terms 6x and -2x on the left-hand side:\[4x + 8 = 24\]

- Isolate the variable term

Subtract 8 from both sides to isolate the term with the variable:\[4x + 8 - 8 = 24 - 8\]which simplifies to \[4x = 16\]

- Solve for x

Finally, divide both sides by 4 to solve for x: \[x = \frac{16}{4}\]which simplifies to \[x = 4\]

Key concepts

Distributive Property

The Distributive Property is a key concept in algebra. It allows you to multiply a number by a group of numbers added together within parentheses. In the example from the exercise, we had the equation \(6x - 2(x - 4) = 24\).
The term \(- 2(x - 4)\) means we need to distribute or multiply \(-2\) by both \(x\) and \(-4\)..
By applying this property, the equation transforms as follows:
\[-2(x - 4) = -2 \times x + (-2) \times (-4) = -2x + 8\]. This conversion simplifies our equation to \(6x - 2x + 8 = 24\).
Understanding the Distributive Property helps you simplify and solve more complex equations.

Combining Like Terms

Combining Like Terms involves simplifying expressions by adding or subtracting variables of the same type. In the equation from the exercise, after applying the distributive property, we had \(6x - 2x + 8 = 24\).
Here, \(6x\) and \(-2x\) are 'like terms' because both contain the variable \(x\).
Combining them results in \(6x - 2x = 4x\), simplifying our equation to \(4x + 8 = 24\).
This step reduces the complexity of the equation, making it easier to solve.
Always look for terms with the same variable and combine them to simplify your equations quickly.

Isolating Variable

Isolating the variable means getting the variable term alone on one side of the equation. This is crucial for solving equations. In our example, the equation was \(4x + 8 = 24\) after combining like terms.
To isolate \(4x\), you need to remove \(+8\) from the left side by subtracting \(-8\) from both sides of the equation:
\[4x + 8 - 8 = 24 - 8\].
This simplifies to \(4x = 16\).
Isolating the variable helps you clearly see the next steps needed to solve for the variable.

Solving for Variable

Once the variable is isolated, solving for it is straightforward. With the equation \(4x = 16\) from our previous step, the only task left is to solve for \(x\).
To do this, divide both sides by 4:
\[4x/4 = 16/4\].
This simplifies to \(x = 4\).
Solving for a variable essentially means finding the value of the variable that makes the equation true.
This final step confirms that your earlier steps were correct, and you've successfully solved the equation.