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Understanding algebraic expressions is a fundamental concept in algebra. An algebraic expression consists of numbers, variables, and operation symbols that are put together to represent a specific value or relationship. For example, in the given exercise, the statement 'Emilio has three fewer than double the number Jacob has' can be represented by the expression \(2x - 3\). Here, \(2x\) means twice Jacob's coins, and subtracting 3 makes it 'three fewer.'
An algebraic expression does not have an equal sign, distinguishing it from an equation. When creating expressions, it's crucial to correctly interpret the words and convert them into mathematical symbols.
Practice creating algebraic expressions from different word problems to gain confidence!

Variables play a vital role in algebra and problem-solving. A variable is a symbol, often a letter, that represents one or more numbers. In our exercise, the variable \(x\) stands for the number of coins Jacob has. By defining \(x\), we can easily describe quantities that depend on Jacob's coins using algebraic expressions.
Utilizing variables helps simplify complex problems by turning word problems into manageable mathematical equations. For instance, since Jacob's coins are \(x\), Emilio's and Latisha's coins can be expressed in terms of \(x\) (i.e., \(2x - 3\) and \(3x + 20\), respectively).
Always start by defining variables clearly to ensure the equations you set up correctly model the given problem.

Simplifying equations is a process used to make solving them easier. This involves combining like terms and reducing equations to their simplest form. In our example, we start with the equation: \[ x + (2x - 3) + (3x + 20) = 65 \] Combining like terms, we add the coefficients of \(x\) and the constant numbers separately: \[ x + 2x + 3x - 3 + 20 = 65 \] Simplifying further, this reduces to \[ 6x + 17 = 65 \] From here, we isolate \(x\) by following appropriate algebraic steps such as subtraction and division.
Clear and methodical simplification ensures that each step is correct and helps prevent errors in the final solution. Practice simplifying different equations to become proficient at this skill.

Effective problem-solving in algebra requires systematic steps. Here’s how we solved the given problem:

• Define a variable: Let \(x\) be the number of coins Jacob has.
• Express related quantities in terms of \(x\): Emilio has \(2x - 3\), and Latisha has \(3x + 20\).
• Set up an equation: Sum up all our expressions and set it equal to the total number of coins, 65.
• Simplify the equation: Combine like terms and reduce to a simpler form.
• Solve for the variable: Isolate \(x\) and find its value.
• Substitute back to find specific amounts: Calculate each friend’s coins using the solution for \(x\).

Following these structured steps helps break down a problem into manageable parts, allowing for a clear path to the solution. Practice these steps with various problems to develop a strong problem-solving method.