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Chapter 1: Problem 15

Solve each equation. $$3 x+5-5(x+1)=6 x+7$$

Short Answer

Expert verified
x = -\frac{7}{8}

Step by step solution

01

Distribute and Simplify

Apply the distributive property to eliminate the parentheses. Rewrite the equation: \[3x + 5 - 5(x + 1) = 6x + 7\].Distribute \(-5\) through \((x + 1)\):\[3x + 5 - 5x - 5 = 6x + 7\].
02

Combine Like Terms

Combine like terms on the left-hand side \[3x - 5x + 5 - 5 = 6x + 7\].This simplifies to:\[-2x = 6x + 7\].
03

Isolate Variable Terms on One Side

Move \(6x\) to the left side of the equation by subtracting \(6x\) from both sides:\[-2x - 6x = 7\].This simplifies to:\[-8x = 7\].
04

Solve for x

Divide both sides by \(-8\) to solve for \(x\):\[x = \frac{7}{-8}\].This simplifies to:\[x = -\frac{7}{8}\].

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Key Concepts

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

Distributive Property
To solve linear equations, we often start by applying the distributive property. This property allows us to remove parentheses by distributing a factor outside the parentheses to each term inside them.
For example, consider the equation: \[3x + 5 - 5(x + 1) = 6x + 7.\]
First, we distribute the \(-5\) through \(x + 1\). This means we multiply \(-5\) by both \(x\) and \(1\).
So, \( -5(x + 1) = - 5x - 5 \).
Applying this to our equation, we get: \[3x + 5 - 5x - 5 = 6x + 7.\]
Distributing helps simplify the equations and makes it easier to solve for the variable.
Combining Like Terms
After using the distributive property, the next step is to combine like terms. Like terms are terms that contain the same variables raised to the same power.
In our simplified equation \[3x + 5 - 5x - 5 = 6x + 7,\] the like terms on the left-hand side are \(3x\) and \(-5x\). Similarly, \(+5\) and \(-5\) are constant terms.
Combining these like terms gives us: \[3x - 5x + 5 - 5 = -2x.\]
Combining like terms simplifies the equation further and helps to isolate the variable.
Isolating Variables
The final key step is isolating the variable to one side of the equation. This makes it possible to solve the equation.
In our equation \(-2x = 6x + 7\), we need to get all \(x\)-terms on one side. To do this, we subtract \(6x\) from both sides: \[-2x - 6x = 7.\]
This simplifies to: \[-8x = 7.\]
Finally, to solve for \(x\), we divide both sides by \(-8\): \[x = \frac{7}{-8}.\]
This simplifies to: \[x = -\frac{7}{8}.\]
By isolating the variable, we find the solution to the equation.

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