Problem 20 - Algebra 1

Question

Solve the equation. $$30=16+\frac{1}{5} x$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 70\).

Step 1: Isolate the term with variable \(x\)

In order to isolate \(x\), first subtract 16 from both sides of the equation. This results in \(30 - 16 = 16 - 16 + \frac{1}{5} x\), simplifying this gives \(14 = \frac{1}{5} x\).

Step 2: Solve for \(x\)

To further isolate \(x\), multiply both sides of the equation by 5 (as it is the denominator of the fraction with \(x\)). This gives \(14 * 5 = \frac{1}{5} x * 5\), simplifying this gives \(70 = x\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is all about rearranging equations to make them easier to solve. In the context of solving linear equations, we often need to isolate the variable—we want it by itself on one side of the equation.

This involves a series of operations that simplify the equation:

• Addition/Subtraction: If there's a number on the side with the variable, we use addition or subtraction to remove it. For instance, in the equation \(30 = 16 + \frac{1}{5}x\), we subtract 16 from both sides to simplify it to \(14 = \frac{1}{5}x\).
• Multiplication/Division: Once the variable is isolated with a coefficient or fraction, you may need to multiply or divide both sides to solve for the variable. This helps you to "undo" the operations around the variable.

Algebraic manipulation requires understanding the equality property, which states that doing the same operation to both sides of an equation keeps the equation balanced.

Fractions in Equations

Fractions can seem daunting in algebra, but they are just numbers that can be handled with the same principles. When you encounter an equation with a fraction, like \(14 = \frac{1}{5} x\), treat the fraction as part of the operations involving the variable.

Here are some steps to tackle fractions in equations:

• Understand the Fraction: Identify the numerator and denominator. In \(\frac{1}{5}x\), '1' is the numerator, and '5' is the denominator.
• Eliminate the Fraction: To eliminate the fraction, multiply both sides of the equation by the denominator. This helps you get rid of the division implied by the denominator.
• For \(14 = \frac{1}{5} x\), multiplying both sides by '5' removes the fraction, resulting in \(70 = x\).

Treat fractions step by step, and don't rush through it. Taking it slow ensures no mistakes are made in balancing your equation.

Distributive Property

The distributive property is a blueprint for breaking down expressions into simpler components. It expands expressions that involve multiplication across terms inside parentheses. While the given equation doesn't have a parent's terms that expressly require distribution, understanding this property can be helpful.

Imagine you have an expression like \(a(b + c)\), it can be rewritten using the distributive property as \(ab + ac\).

• Key Point: The distributive property doesn't just distribute numbers but also operations across a sum or difference inside parentheses.
• A practical example in algebra simplifies computation, especially in longer, more complex equations.
• Application: If future exercises involved an expression like \(5(x + \frac{1}{5})\), using distributive property: \(5x + 1\).

Grasping the distributive property now will aid in solving a wide array of algebraic equations as you progress.