Question

Solve the equation $$ 4(2-n)=1 $$

Short answer

The solution to the equation is \(n = 1.75\).

Step-by-step solution

Distribute 4

Start by distributing the 4 in the expression, simplify the left side of the equation \(4(2-n) = 1\) to obtain \(8 - 4n = 1\).

Rearrange the equation

Then, isolate the term involving \(n\) by subtracting 8 from both sides of the equation. This yields \(-4n = 1 - 8\). Simplify the right side to give \(-4n = -7\).

Solve for n

Lastly, solve for \(n\) by dividing both sides of the equation by -4. This gives \(n = -7 / -4\), which simplifies to \(n = 1.75\).

Key concepts

Distributive Property

Understanding the distributive property is key to solving equations like the given problem, where a number is multiplied by a parenthesis. In essence, the distributive property allows you to remove the parenthesis by multiplying each term inside the parenthesis by the number outside.

For our example, applying the distributive property looks like this: you take the number 4 and multiply it by each term inside the parenthesis (2 and -n), which gives us two new terms, 8 and -4n. Therefore, the expression transforms as follows: from the original form of \( 4(2-n) \) to the distributed form \( 8 - 4n \). This step is crucial because it simplifies the equation and sets the stage for further steps.

Isolating Variables

The next step in solving linear equations is to isolate the variable, which means to get the variable on one side of the equation and the numbers on the other. This process makes it easier to find the value of the variable.

To isolate \( n \) in our equation \( 8 - 4n = 1 \), you need to move the constant (8) to the other side. This is done by subtracting 8 from both sides, leading to \( -4n = -7 \). It's a bit like balancing scales; whatever you do to one side, you must do to the other to maintain the balance. Isolating the variable sets us up to find its value in the final step.

Equation Simplification

The final piece of the puzzle is equation simplification. After isolating the variable, you're left with an equation that might still need to be simplified to find the solution. As with our equation \( -4n = -7 \), we simplify further by dividing both sides by -4 to solve for \( n \).

Simplification involves performing basic arithmetic to achieve the most reduced form of the equation possible. Here, \( n = -7 / -4 \) simplifies to \( n = 1.75 \). It's like peeling away all the extra details until you're left with just the answer you're looking for. Simplification is the final step that yields the solution to the equation.