Problem 34 - Algebra for College Students

Question

Solve each equation. \(-5(x-1)=12\)

Step-by-Step Solution

Answer

The solution is \(x = -\frac{7}{5}\).

Step 1: Distribute the -5

Apply the distributive property to the equation to remove the parentheses. Multiply -5 by each term inside the parenthesis: \(-5(x-1) = -5 \times x + (-5) \times (-1)\),which simplifies to:\[-5x + 5 = 12\].

Step 2: Isolate the Variable Term

Subtract 5 from both sides of the equation to move the constant term to the right side. This produces:\[-5x = 12 - 5\],which simplifies to:\[-5x = 7\].

Step 3: Solve for x

Divide both sides by -5 to solve for \(x\): \[x = \frac{7}{-5}\], which further simplifies to:\[x = -\frac{7}{5}\].

Key Concepts

Distributive Property

The distributive property is a handy tool in algebra and is often the first step in solving equations that include parentheses. It's like spreading out the numbers when multiplying across a summation or subtraction. When you come across an equation such as \(-5(x-1) = 12\), you need to distribute the number outside the parenthesis to each term inside. Here, \(-5\) is multiplied by \(x\) and \(-1\).

• \(-5 \times x = -5x\)
• \(-5 \times (-1) = 5\)

This results in: \(-5x + 5 = 12\).
Using the distributive property effectively transforms complex equations into simpler forms, helping you to eliminate those pesky parentheses right away. Don't forget to watch your signs when doing your multiplication!

Isolating Variable

Isolating the variable means moving everything else to one side of the equation so that your variable is all by itself on the other side. This is like de-cluttering a pile so you can see exactly what you're left to work with. Once we have the equation \(-5x + 5 = 12\), our job is to get \(-5x\) all alone on one side.
To achieve this, subtract \(5\) from both sides. Simplifying that, our equation becomes: \(-5x = 12 - 5\) which is \(-5x = 7\).
This action of isolating transforms our equation into something much simpler, setting the stage for the final solution step. Remember, the goal here is to have the term with the variable on one side and the constant terms shifted to the other side.

Equation Simplification

Simplifying the equation involves doing the math required to make things less... well, complicated. Once you've isolated the variable term, like our \(-5x = 7\), it's time to solve for \(x\).
To simplify, divide each side of the equation by \(-5\), the coefficient of \(x\). This operation gives you:\[x = \frac{7}{-5}\] which simplifies to:\[x = -\frac{7}{5}\].
Equation simplification focuses on breaking steps down to handle any pesky fractions or decimals. Always check that your variable is truly isolated and your equation reduced to its most understandable form. Keep it straightforward so you can easily see the solution!