Problem 35 Solve each equation: $$ -3(3... [FREE SOLUTION]

Chapter 7: Problem 35

Solve each equation: $$ -3(3 x-1)=2 $$

Short Answer

Expert verified

The solution is $ x = \frac{1}{9} $.

Step by step solution

Distribute the Constants

First, distribute the $ -3 $ across the parenthesis $ (3x - 1) $. This results in \[ -3 \cdot 3x + (-3) \cdot (-1) = 2 \]. Calculate the distribution as follows: \[ -9x + 3 = 2 \].

Isolate the Variable Term

Next, move the constant term $ +3 $ to the other side of the equation by subtracting $ 3 $ from both sides. This gives: \[ -9x + 3 - 3 = 2 - 3 \]. Simplifying, we get: \[ -9x = -1 \].

Solve for the Variable

Now, divide both sides of the equation by $ -9 $ to solve for $ x $. This results in: \[ x = \frac{-1}{-9} \]. Simplifying the fraction, we have: \[ x = \frac{1}{9} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property

The distributive property is a fundamental principle in mathematics used to simplify expressions. It involves distributing a factor over terms inside parentheses. In the equation $-3(3x-1)=2$, we apply the distributive property to eliminate the parentheses. Here鈥檚 how it works:

Multiply $-3$ by each term inside the parentheses: $3x$ and $-1$.
This operation results in $-3 \, \cdot \, 3x + (-3) \, \cdot \, (-1)$, which simplifies to $-9x + 3$.

This process is essential because it transitions the equation from a more complex form into a linear form that鈥檚 easier to solve. Remember to preserve the sign in front of each term as you multiply. This initial step sets the stage for isolating the variable in subsequent steps.

Isolating the Variable

Isolating the variable is all about getting the variable term on one side of the equation. This is crucial for solving the equation. After applying the distributive property, you should focus on moving all other terms to the opposite side. In our example, we have:

An equation after distribution: $-9x + 3 = 2$.
To isolate $x$, subtract $3$ from both sides to eliminate it from the left side.
This gives $-9x + 3 - 3 = 2 - 3$, leading to $-9x = -1$.

The key here is to perform the same operation on both sides of the equation to maintain balance. The goal is to simplify until $x$ or the variable of interest is by itself on one side. Successfully isolating the variable helps to determine its value in the final step.

Solving for x

In the final step of solving a linear equation, we find the exact value of the variable, in this instance, $x$. Once the variable is isolated, we solve it by performing basic arithmetic operations. Here鈥檚 what we do:

Our isolated equation is $-9x = -1$.
To solve for $x$, divide both sides by $-9$, which is the coefficient of $x$.
This yields $x = \frac{-1}{-9}$.
Simplifying gives $x = \frac{1}{9}$.

Remember, dividing by a negative changes the sign of the result. Solving for $x$ completes the equation as it provides the value needed to satisfy the original equation. This solution technique is crucial for understanding how different algebraic equations work and finding variable values effectively.

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91影视

Solve each equation: $$ -3(3 x-1)=2 $$

Short Answer

Step by step solution

Distribute the Constants

Isolate the Variable Term

Solve for the Variable

Key Concepts

Distributive Property

Isolating the Variable

Solving for x

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