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Chapter 7: Problem 11

The product of 8 , and a number increased by 6 , is 104 . What is the number?

Short Answer

Expert verified
The number is 7.

Step by step solution

01

- Formulate the equation

Let the unknown number be represented by the variable \( x \). According to the problem, the product of 8 and the number increased by 6 is equal to 104. This can be written as: \[ 8(x + 6) = 104 \]
02

- Simplify the equation

Distribute the 8 through the parentheses: \[ 8x + 48 = 104 \]
03

- Isolate the variable

Subtract 48 from both sides of the equation to isolate the term with the variable: \[ 8x = 104 - 48 \] This simplifies to: \[ 8x = 56 \]
04

- Solve for the variable

Divide both sides of the equation by 8 to solve for \( x \): \[ x = \frac{56}{8} \] Therefore, \[ x = 7 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a key concept when working with algebraic equations. It tells us that multiplying a number by a sum is the same as doing each multiplication separately. For example, in the equation \[ 8(x + 6) = 104 \] We need to distribute the 8 to both terms inside the parentheses: \[ 8 \times x + 8 \times 6 = 8x + 48 \] By distributing 8 through the parentheses, we simplify the expression before moving on to isolating the variable. Understanding how to correctly apply this property is crucial for solving more complex equations.
Isolating Variables
Once we've simplified our equation using the distributive property, the next step is to isolate the variable. This means getting the variable by itself on one side of the equation. In our example: \[ 8x + 48 = 104 \] We need to move the 48 to the other side to isolate the term containing the variable. We do this by subtracting 48 from both sides: \[ 8x = 104 - 48 \] This simplifies to: \[ 8x = 56 \] By isolating the variable, we make solving the equation much easier.
Simplifying Equations
Simplifying equations involves performing operations to boil an equation down to its simplest form. It makes further steps more straightforward and manageable. In our example, after isolating the term with the variable, our simplified equation is: \[ 8x = 56 \] To solve for \( x \), we need to eliminate the coefficient (the number multiplying the variable). We do this by dividing both sides of the equation by 8: \[ x = \frac{56}{8} \] This calculation gives us: \[ x = 7 \] By systematically simplifying the equation, we find that the unknown number is 7.

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Most popular questions from this chapter

Solve each system by the elimination method. Check each solution. $$ \begin{array}{l} 5 x+8 y=10 \\ 24 y=-15 x-10 \end{array} $$

A motor scooter travels \(20 \mathrm{mi}\) in the same time that a bicycle travels \(8 \mathrm{mi}\). If the rate of the scooter is 5 mph more than twice the rate of the bicycle, find both rates.

Solve each system by graphing. If the system is inconsistent or the equations are dependent, say so. \(2 x=-3 y\) \(4 x-y=0\)

Solve each system by graphing. If the system is inconsistent or the equations are dependent, say so. \(4 x+y=5\) \(y=\frac{3}{2} x-6\)

Without graphing, do the following for each system of equations. (a) Describe each system. (b) State the number of solutions. (c) Is the system inconsistent, are the equations dependent, or neither? \(3 x+2 y=5\) \(6 x+4 y=3\)
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