Problem 17 - Algebra for College Students

Question

Translate each verbal sentence into an equation, using \(x\) as the variable. Then solve the equation. If the product of a number and -4 is subtracted from the number, the result is 9 more than the number. Find the number.

Step-by-Step Solution

Answer

The number is 2.25.

Step 1 - Set Up the Equation

Translate the verbal sentence into an equation. Let's call the unknown number \(x\). According to the problem, the product of the number and -4 (which is \( -4x \)) is subtracted from the number (which is \( x \)). This result is 9 more than the number. So the equation is: \[ x - (-4x) = x + 9. \]

Step 2 - Simplify the Equation

Simplify the left side of the equation: \[ x - (-4x) \ = x + 4x \ = 5x. \] So the equation becomes: \[ 5x = x + 9. \]

Step 3 - Solve for x

Subtract \( x \) from both sides to get the \( x \) terms on one side of the equation: \[ 5x - x = x + 9 - x, \] which simplifies to: \[ 4x = 9. \] Finally, divide both sides by 4: \[ x = \frac{9}{4} = 2.25. \]

Key Concepts

Verbal to Algebraic Translation

The first step in solving word problems in mathematics is to translate words into algebraic expressions. This process might seem tricky at first, but it's important to identify the key elements and their relationships. In our exercise, we started with a verbal sentence about a number. We called this unknown number x. The sentence, 'the product of a number and -4 is subtracted from the number,' means we first multiply x by -4, resulting in -4x, and then subtract this from x. The phrase 'the result is 9 more than the number' indicates that one side of the equation will be x + 9. Combining these pieces, the verbal sentence becomes the algebraic equation: \[ x - (-4x) = x + 9. \]
This translation is crucial because it turns a written problem into a mathematical one, which we can then solve systematically.

Simplifying Equations

So, the equation simplifies to:\[ 5x = x + 9. \]
This step helps to simplify the complex-looking equation into a simpler form, focusing on combining like terms when possible.

Steps in Problem-Solving

Having simplified the equation, we proceed to the final steps to find the solution. Solving equations is often done through the following steps:

• Getting all the variable terms on one side of the equation,
• Isolating the variable,
• Solving for the variable.

For our equation, \[ 5x = x + 9, \]
we first subtract x from both sides to get all the x terms on one side: \[ 5x - x = x + 9 - x, \] which simplifies to \[ 4x = 9. \]
Next, we isolate x by dividing both sides by 4:\[ x = \frac{9}{4} = 2.25. \]
We find that x equals 2.25. By following these structured steps, you can handle complex equations more efficiently and accurately. Remember, practice makes perfect!