Problem 24 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: Combine Like Terms

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

Step 4: 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.

Step 5: 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.

Key Concepts

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’s 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.