Question

Solve each problem for the required quantity. Four less than 6 times a number is \(32 .\) Find the number.

Short answer

The number is 6.

Step-by-step solution

Translate the problem into an equation

Translate the words into a mathematical equation. 'Four less than 6 times a number is 32' can be written as \(6x - 4 = 32\), where \(x\) is the unknown number we are trying to find.

Isolate the variable

Add 4 to both sides of the equation to isolate the term containing the variable on one side. This gives us \(6x - 4 + 4 = 32 + 4\), which simplifies to \(6x = 36\).

Solve for the variable

Divide both sides of the equation by 6 to solve for \(x\). This gives us \(x = \frac{36}{6}\), so \(x = 6\).

Key concepts

Translating Verbal Expressions into Equations

Understanding how to convert verbal statements into algebraic expressions is a foundational skill in algebra. It involves identifying the numerical and variable components described in the words and representing them using mathematical symbols.

When engaging in translating verbal expressions, pay careful attention to phrases that indicate operations: 'less than' suggests subtraction, 'times' indicates multiplication, and phrases like 'is equal to' point to the equality sign. For example, the phrase 'four less than 6 times a number' would be expressed as a subtraction ('less than') after a multiplication ('times a number'). The unknown number we refer to in algebra is typically represented by a variable, like 'x'.

Thus, translating the phrase into an equation would yield the algebraic expression \(6x - 4 \), with '6 times a number' becoming \(6x \) and 'four less than' informing the subtraction of 4.

Isolating the Variable

The next critical step in solving algebraic equations is to isolate the variable on one side of the equation. The goal is to get the variable by itself so that what remains is the solution to the equation.

Isolating the variable typically requires using inverse operations. If a term is subtracted from the variable, we add the same number to both sides of the equation. Similarly, if the variable term is multiplied by a number, dividing both sides by that number will isolate the variable.

Using the example \(6x - 4 = 32\), to isolate \(x\), we add 4 to both sides of the equation. This cancels the -4 on the left and adds 4 to 32 on the right, effectively moving the -4 across the equality and changing the equation to \(6x = 36\).

Solving Linear Equations

Once a variable is isolated, solving linear equations becomes more straightforward. In our current example, we have arrived at the equation \(6x = 36\). A linear equation is one where the variable is raised to the power of 1 and hence plots as a straight line on a graph.

To solve for \(x\), we perform the operation that will leave \(x\) on one side of the equation and a numerical value on the other. Since \(x\) is currently being multiplied by 6, we divide both sides by 6 to undo this multiplication, which gives us \(x = \frac{36}{6}\), simplifying further to \(x = 6\).

It's important when solving these equations to perform the same operation on both sides, maintaining the balance of the equation. This ensures the integrity of the equation is preserved and the correct solution is found.