Problem 37 Solve each equation. $$ (5 x... [FREE SOLUTION]

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Chapter 5: Problem 37

Solve each equation. $$ (5 x+1)(x+3)=-2(5 x+1) $$

Short Answer

Expert verified

x = -5

Step by step solution

- Identify the common factor

Look for terms common on both sides of the equation. Notice that $5x + 1$ appears on both sides of the equation.

- Simplify the equation

Since $5x + 1$ is a common factor, we can divide both sides of the equation by $5x + 1$, assuming $5x + 1 \eq 0$. This gives: $ x + 3 = -2 $.

- Solve for x

Now, solve the simplified equation. \ \[ \begin{aligned} \ x + 3 &= -2 \ \ x &= -2 - 3 \ \ x &= -5 \end{aligned} \]

- Check the initial condition

Also check the initial condition $5x + 1 \eq 0$. For $x = -5 \Rightarrow 5(-5) + 1 = -25 + 1 = -24 \eq 0$. Hence, $x = -5$ is a valid solution.

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Key Concepts

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

Common Factor

When solving linear equations, identifying a common factor can simplify the process. A common factor is a term that appears in multiple parts of an equation. For example: in the equation $ (5x + 1)(x + 3) = -2(5x + 1) $, $5x + 1 $ is a common factor on both sides. By spotting this, we can make the problem easier to handle. Dividing both sides by this common factor (assuming it's not equal to zero) streamlines the equation, allowing us to focus on the remaining terms. This step not only simplifies the problem but also helps avoid unnecessary complication. Always watch out for common factors to make progress quickly.

Simplification

Simplification is a crucial step in solving equations. Once you identify a common factor, simplifying the equation becomes straightforward. In our example, after recognizing that $5x + 1 $ appears on both sides, we can divide each side by $5x + 1 $, resulting in: $ x + 3 = -2 $. Simplifying equations often involves canceling out terms, combining like terms, or dividing by common factors. By breaking down complex expressions into simpler forms, you can make the problem more approachable and easier to solve. Remember, the goal is to transform the equation to its simplest form possible before isolating the variable.

Solving Equations

Solving equations involves isolating the variable to find its value. After simplifying the equation $ x + 3 = -2 $, the next step is to solve for $ x $. This means getting $ x $ by itself on one side of the equation: \[ x = -2 - 3 \] \[ x = -5 \]. After finding $ x $, always check your solution by plugging it back into the original equation to verify it satisfies all conditions. In our case: $ 5(-5) + 1 = -25 + 1 eq 0 $, confirming $ x = -5 $ is valid. This process ensures that no steps were missed and the solution holds true for the given equation.

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91影视

Solve each equation. $$ (5 x+1)(x+3)=-2(5 x+1) $$

Short Answer

Step by step solution

- Identify the common factor

- Simplify the equation

- Solve for x

- Check the initial condition

Key Concepts

Common Factor

Simplification

Solving Equations

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