Question

Solve the equation. $$ 5(w-5)=25 $$

Short answer

The solution to the equation is w = 10.

Step-by-step solution

Distribute

Distribute 5 inside the parentheses: 5(w-5) = 5w - 25. The equation is now 5w - 25 = 25.

Add 25 to both sides

To isolate the 5w on one side, add 25 to both sides of the equation: 5w - 25 + 25 = 25 + 25. The equation simplifies to 5w = 50.

Divide both sides by 5

To get w by itself, divide both sides of the equation by 5: \[ \frac{5w}{5} = \frac{50}{5} \]. The solution of the equation is w = 10.

Key concepts

Distributive Property

The distributive property is a foundational concept in algebra that allows us to multiply a single term by each term inside a parenthesis. For instance, consider the equation from the exercise:
\[ 5(w-5)=25 \].
To apply the distributive property, we would multiply 5 by both 'w' and '-5', giving us
\[5 \times w - 5 \times 5\].
The result would simplify to
\[ 5w - 25 \].
This step is crucial as it transforms the equation into a form where the variable can be isolated, which is the next important step in solving the equation. When using the distributive property, it's best to proceed methodically, ensuring that each term inside the parenthesis is multiplied by the term outside, and to double-check your work to avoid simple arithmetic errors.

Isolate the Variable

Once the distributive property has been applied, the next goal in solving linear equations is to isolate the variable. Isolating the variable means to get the variable on one side of the equation by itself. In our example, after using the distributive property, we had:
\[ 5w - 25 = 25 \].
To isolate the variable 'w', we need to remove the '-25' from the left side. We do this by adding 25 to both sides of the equation:
\[ 5w - 25 + 25 = 25 + 25 \].
This gives us the simplified equation
\[ 5w = 50 \].
At this stage, 'w' is not yet completely isolated because it is being multiplied by 5. To fully isolate 'w', we divide both sides by 5, which gives us the final solution
\[ w = 10 \].
This step of isolating the variable is crucial as it reveals the value of the variable that solves the equation. Remember, balancing the equation by performing the same operation on both sides maintains the equality and ensures a correct solution.

Equation Simplification

Simplifying an equation makes it easier to solve. After applying the distributive property and working towards isolating the variable, we often perform simplification steps. Equation simplification involves combining like terms and reducing expressions to their simplest form. In our example, we simplified by adding 25 to both sides to cancel out the minus 25 on the left. This step resulted in a cleaner and more manageable equation:
\[ 5w = 50 \].
Furthermore, by dividing both sides by 5, we simplify the equation to find the value of 'w'. Simplification helps in minimizing errors and ensuring that solving the equation can be done in a more straightforward and understandable manner. Always check to simplify at every stage of solving an equation, such as reducing fractions or combining like terms, as this keeps the solution process clean and logical.