Problem 23 - Algebra for College Students

Question

Determine whether each is an expression or an equation. Simplify any expressions, and solve any equations. $$ 5(x+3)-8(2 x-6)=12 $$

Step-by-Step Solution

Answer

This is an equation. Solving for x, the value is x = 51 / 11.

Step 1: Identify if it's an expression or an equation

The given problem is: 5(x+3)-8(2x-6)=12 An equation has an equality sign \(=\). Since we have the equality sign, this is an equation.

Step 2: Distribute the constants inside the parentheses

Distribute the constants 5 and -8 into the terms inside the parentheses: 5(x+3) = 5x + 15 -8(2x-6) = -16x + 48 The equation then becomes: 5x + 15 - 16x + 48 = 12.

Step 3: Combine like terms

Combine the terms involving x and the constant terms: 5x - 16x + 15 + 48 = 12 Combine them to get: -11x + 63 = 12.

Step 4: Isolate the variable term on one side

Subtract 63 from both sides to isolate the term with the variable (x): -11x + 63 - 63 = 12 - 63 This simplifies to: -11x = -51.

Step 5: Solve for the variable

Divide both sides by -11 to solve for x: \(-11x / -11 = -51 / -11\) This simplifies to: x = 51 / 11.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill in algebra. It involves combining like terms, which are terms that have the same variables raised to the same power. This helps in reducing the expression to its simplest form.
In our exercise, we start by simplifying the terms inside the parentheses. For the given equation, we have 5(x+3) and -8(2x-6). Use the distributive property to expand these terms:
5(x+3) = 5x + 15
-8(2x-6) = -16x + 48
This gives a more straightforward form of the equation: 5x + 15 - 16x + 48 = 12. Next, combine the like terms (terms with x and constant terms):
5x and -16x are like terms, so combine them: 5x - 16x = -11x.
Also, 15 and 48 are constants, so sum them up: 15 + 48 = 63.
Now we have a simplified equation: -11x + 63 = 12.

Distributive Property

The distributive property is a useful tool in algebra that allows you to multiply a single term by each term within parentheses. It helps simplify expressions and is crucial for solving equations. The distributive property can be written as:
\[a(b + c) = ab + ac\]

In our given equation, we used the distributive property twice:
5(x + 3) = 5 \times x + 5 \times 3 = 5x + 15
and
-8(2x - 6) = -8 \times 2x + (-8) \times (-6) = -16x + 48.

This transformation helps in laying down the equation in a simpler form to solve for the variable x. Always remember to multiply each term inside the parentheses individually by the term outside.

Isolating Variables

Solving equations often involves isolating the variable on one side of the equation. This means you want to get all terms with the variable on one side and the constants on the other side.

• Start by combining like terms as seen in the previous sections.
• Next, move the constant terms to the opposite side of the equation from the variable terms.

For our specific example: We start from -11x + 63 = 12.
To isolate the variable term (-11x), subtract 63 from both sides:
-11x + 63 - 63 = 12 - 63
This simplifies to: -11x = -51.
Finally, solve for x by dividing both sides by -11: \ \frac{-51}{-11} = x \
This simplifies to:
\[x = 51 / 11\].
Now you've isolated and solved for the variable x. Practice makes this process more intuitive!