Problem 45

Question

Solve the equation. $$-2(7-5 x)=10$$

Step-by-Step Solution

Verified

Answer

The solution of the equation -2(7-5x)=10 is $x = 2.4$.

1Step 1: Apply Distributive Property

Applying the distributive property of multiplication over subtraction to the left side of the equation gives $-2 * 7 + (-2) * -5x = -14 + 10x$. So the original equation $-2(7-5x)=10$ simplifies to $-14 + 10x = 10$.

2Step 2: Simplify The Equation

Next, rearrange the equation to have the $x$ term on one side and the numbers on the other side. To do that, add 14 to both sides to get rid of -14 on the left side. That gives $10x = 10 + 14$, which simplifies to $10x = 24$.

3Step 3: Solve For The Variable

Finally, divide both sides of the equation by 10 to solve for $x$. Thus, the solution is $x = \frac{24}{10}$ or $x = 2.4$.

Key Concepts

Distributive PropertySimplifying EquationsVariable Isolation

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions and equations. When you have an expression like $-2(7 - 5x)$, the distributive property allows you to multiply each term inside the parentheses by the number outside. Here's how it works:

Multiply $-2$ by $7$, resulting in $-14$.
Multiply $-2$ by $-5x$, which gives $+10x$. Remember that multiplying two negative numbers results in a positive number.

Combining these results, the expression $-2(7 - 5x)$ becomes $-14 + 10x$. This property is especially useful when beginning to solve linear equations, as it allows you to break down complex problems into simpler parts.
As you practice, spotting where to apply the distributive property will become a fast and intuitive step in solving equations.

Simplifying Equations

Simplifying equations often involves rearranging terms to make them easier to work with. In the context of the original problem, simplifying the equation means getting all terms involving the variable $x$ on one side and the constant numbers on the other side.
Let's look at how this process occurs step-by-step:

First, you apply the distributive property to change $-2(7 - 5x)$ to $-14 + 10x$.
Next, to isolate the term $10x$, you need to get rid of $-14$ from that side of the equation.

To do this, add $14$ to both sides, resulting in an equation like $10x = 10 + 14$. Why add 14? The property of equality states that what you do to one side of the equation must be done to the other, preserving balance. This step-by-step simplification process paves the way for the final isolation of the variable.

Variable Isolation

Variable isolation is the process of solving an equation by getting the variable—let’s say $x$—alone on one side of the equation. After simplifying an equation, like we did to reach $10x = 24$, isolation means making $x$ the subject.
Here's how to isolate $x$:

You have $10x = 24$. Your next step is to divide each side by $10$, the coefficient of $x$.
This division simplifies the equation to $x = \frac{24}{10}$.
Further simplify the fraction: $x = 2.4$.

By isolating $x$, you find the value that satisfies the equation. This method of isolation is a key procedure in algebra and helps in solving for various unknowns in different equations. Understanding and mastering these steps will make tackling even more complex equations easier in the future.

Problem 45

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