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Chapter 1: Problem 24

Determine whether each is an expression or an equation. Simplify any expressions, and solve any equations. $$ -7(x+4)+13(x-6)=18 $$

Short Answer

Expert verified
It is an equation. The value of x is 62/3 or approximately 20.67.

Step by step solution

01

Identify the Problem Type

Examine the given mathematical statement -7(x+4)+13(x-6)=18 and determine whether it is an expression or an equation. Since it contains an equals sign '=', it is an equation.
02

Distribute the Constants

Apply the distributive property to both terms on the left side of the equation. This means multiplying -7 by (x+4) and 13 by (x-6). Thus, -7(x + 4) = -7x - 28 13(x - 6) = 13x - 78.
03

Combine Like Terms

Combine all like terms on the left side of the equation: -7x - 28 + 13x - 78 Simplify to get: 6x - 106 = 18.
04

Isolate the Variable Term

Add 106 to both sides of the equation to isolate the term with x. 6x - 106 + 106 = 18 + 106 Simplifies to 6x = 124.
05

Solve for the Variable

Divide both sides by 6 to solve for x x = 124/6 Simplifies to: x = 62/3 or approximately x = 20.67.

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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 one of the fundamental properties in algebra. It allows us to simplify expressions by distributing a single term across terms inside parentheses.
In the given equation, we apply the distributive property to both terms on the left side:
  • For \-7(x + 4)\, multiply -7 by both x and 4.
  • For 13(x - 6), multiply 13 by both x and -6.
By doing this multiplication, the equation \-7(x+4)+13(x-6)=18\ becomes:\[ -7x - 28 + 13x - 78 = 18.\]This step simplifies the expression, making it easier to manage.
Always remember to distribute negative signs as well. It鈥檚 a common mistake to distribute the coefficient but forget the sign.
Combining Like Terms
After applying the distributive property, the equation turns into \( -7x - 28 + 13x - 78 = 18 \).
The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power.
In our equation, the like terms are \-7x\ and \13x\ which both have the variable \x\, and the constants \-28\ and \-78\.
Combine the like terms as follows:
  • \[-7x + 13x = 6x\]
  • \[-28 - 78 = -106\]

This results in the simplified equation: \-106 + 6x = 18.\ Combining like terms helps to reduce the equation into a simpler form, making it easier to solve.
Isolating the Variable
The final core concept is isolating the variable. Here we have the simplified equation: \6x - 106 = 18\.
To isolate \x\, we need it to be by itself on one side of the equation. Initially, we deal with the constant term:
  • Add 106 to both sides of the equation to eliminate the constant term on the left:
    \[6x - 106 + 106 = 18 + 106\]
  • This simplifies to:
    \[6x = 124\]

    • The next step is to isolate \x\ completely by dividing both sides by 6:

  • \[x = \frac{124}{6}\]
    • This evaluates to:
    \[x = \frac{62}{3} \approx 20.67 \]

Isolating the variable is an essential algebraic method because it leads us to the solution of the equation.

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