Problem 46 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 20 x=3 x+17 $$

Step-by-Step Solution

Answer

The solution to the equation \(20x=3x+17\) is \(x=1\).

Step 1: Identify the Equation

The given equation is \(20x=3x+17\). The goal is to isolate \(x\).

Step 2: Apply the Subtraction Property of Equality

Subtract \(3x\) from both sides to consolidate the \(x\) terms on one side. This results in \(20x - 3x = 3x - 3x + 17\), which simplifies to \(17x = 17\).

Step 3: Divide Both Sides by 17

Isolate \(x\) by dividing both sides of the equation by 17 to get \( x = 1 \).

Key Concepts

Equation Solving

Equation solving is the process of finding the value of the variable that makes the equation true. In this context, we are dealing with linear equations, which involve variables raised to the power of one.

An equation is essentially a balance between two expressions separated by an equal sign (=). To solve an equation means to find an unknown variable that balances the equation. For instance, in the equation \(20x = 3x + 17\), the goal is to determine the value of \(x\).

When solving equations, it's crucial to simplify both sides step by step. Recheck your work by substituting the found value back into the original equation to ensure both sides remain equal.

Properties of Equality

The properties of equality are fundamental rules that allow us to manipulate equations without changing their solutions. These properties ensure the balance in equations, critical in solving them.

Some key properties include:

• Addition Property: Adding the same quantity to both sides maintains equality.


• Subtraction Property: Subtracting the same quantity from both sides also maintains equality.


• Multiplication Property: Multiplying both sides by the same nonzero quantity keeps the equation balanced.


• Division Property: Dividing both sides by the same nonzero quantity ensures the equation remains true.

In our example, we used the Subtraction Property of Equality to subtract \(3x\) from both sides, which helps in consolidating the variable terms on one side, simplifying the equation to \(17x = 17\).

Isolating Variables

Isolating variables involves rearranging the equation to get the variable of interest alone on one side of the equation. This step is essential in solving equations, facilitating identification of the unknown value.

In the equation \(20x = 3x + 17\), our target is to get \(x\) by itself. We initially consolidate all \(x\) terms to one side using properties of equality.

Once similar terms are grouped (as in \(17x = 17\)), simplify further by performing operations that leave \(x\) isolated. Here, dividing both sides by 17 was the final maneuver ensuring \(x\) stands alone, resulting in \(x = 1\).

This straightforward method of isolating the variable simplifies calculations and underscores the importance of logical step progression in solving linear equations.