Problem 94 - Introductory Algebra for College Students

Question

In Exercises \(93-96\), let \(x\) represent the number. Express each sentence as a single algebraic expression. Then simplify the expression. Multiply a number by 3. Add 9 to this product. Subtract this sum from the number.

Step-by-Step Solution

Answer

The simplified algebraic expression for the given task is \(-2x - 9\).

Step 1: Translate the textual expressions into mathematical ones

The first instruction from the exercise is to 'Multiply a number by 3'. If 'x' represents the number, this could be expressed as \(3x\). The second instruction is to 'Add 9 to this product', which gives us \(3x + 9\). The final instruction is to 'Subtract this sum from the number', leading to \(x - (3x + 9)\).

Step 2: Simplify the algebraic expression

Now, we simplify the created algebraic expression \(x - (3x + 9)\). After simplifying, by first subtracting \(3x\) from \(x\) and then subtracting 9, we get \(-2x - 9\).

Key Concepts

Translating Word Problems into Algebraic Expressions

Solving algebra problems often starts with turning a word problem into a mathematical one. Firstly, you need to define the variable, like using x to represent an unknown number. Next, identify the operations described in the problem. For example, 'Multiply a number by 3' suggests an operation of multiplication, turning into 3x in algebraic form.

The process requires logical thinking to interpret phrases such as 'Add 9 to this product,' which means to take our previous product 3x and add 9, resulting in 3x + 9. Finally, phrases like 'Subtract this sum from the number' lead to the expression x - (3x + 9). This step-by-step translation is essential to solving the word problem algebraically.

It's important to perform these translations accurately, otherwise the following steps to solve the problem cannot be correctly executed.

Simplifying Algebraic Expressions

After translating the word problem, simplifying expressions is vital. It helps to reduce and condense the expression to something more manageable and easier to work with. To simplify the expression x - (3x + 9), we start by distributing the negative sign into the parenthesis, changing the signs of the terms inside, which results in x - 3x - 9.

Next, we combine like terms, where like terms are terms that have the same variables raised to the same power. Here, x and -3x are like terms, which when combined give us -2x. Now our expression looks like this: -2x - 9. No further simplification can be done because there are no more like terms. Mastery of simplifying expressions makes algebra much more approachable and is a skill that will be used extensively in more complex problems.

Algebraic Manipulation Techniques

Algebraic manipulation involves techniques to restructure algebraic expressions and solve equations. There's a set of rules or operations, known as algebraic properties, that one can apply. These include the distributive property, as we used earlier to simplify the expression x - (3x + 9), and the commutative and associative properties for addition and multiplication.

To deepen your understanding, consider how we could have added an extra step in the problem from the exercise. Before combining like terms, you might rewrite x as 1x to make it more obvious that it is the same type of term as 3x, aiding in simplifying the expression. Other manipulation techniques include factoring, expanding, and using the zero product property to find the roots of equations. Learning and practicing these methods of manipulation can unlock the ability to solve a vast array of algebraic challenges.