Problem 43 - Algebra for College Students

Question

Solve each problem involving consecutive integers. Find four consecutive integers such that the sum of the first three is 54 more than the fourth.

Step-by-Step Solution

Answer

The four consecutive integers are 27, 28, 29, and 30.

Step 1: Identify Consecutive Integers

Let the four consecutive integers be represented by variables. Suppose the smallest integer is denoted by x. Thus, the four consecutive integers are x, x+1, x+2, and x+3.

Step 2: Formulate the Equation

Based on the problem, the sum of the first three integers is equal to 54 more than the fourth integer. Therefore, we can write the equation as follows: x + (x+1) + (x+2) = (x+3) + 54.

Step 3: Simplify the Equation

Combine like terms on both sides of the equation: 3x + 3 = x + 57.

Step 4: Solve for x

Isolate the variable x by subtracting x from both sides and then subtracting 3 from both sides: 3x + 3 - x = x + 57 - x 2x + 3 = 57 2x = 54 x = 27.

Step 5: Determine the Four Consecutive Integers

Now that the value of x has been found, substitute it back in to find the four consecutive integers: x = 27 x+1 = 28 x+2 = 29 x+3 = 30.

Step 6: Verify the Solution

To ensure the solution is correct, verify that the sum of the first three integers is indeed 54 more than the fourth integer: 27 + 28 + 29 = 84 84 - 30 = 54 The solution is verified.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are essential in mathematics. They are statements that show the equality of two expressions. An equation might contain numbers, variables, and operation symbols. For instance, in the exercise provided, the equation is formed as follows:
\[ x + (x+1) + (x+2) = (x+3) + 54 \]
This shows how an equation can represent relationships between consecutive integers.
In algebra, we're often working to find the value of an unknown variable. Here, the variable is x, representing the smallest integer of the four consecutive integers.
Equations help us solve problems by providing a way to model and analyze relationships. Remember, every part of an equation must hold true to maintain the balance. That balance allows us to uncover the value of variables and solve the problem set before us.

Identifying Variables in Math Problems

Variables are symbols used to represent unknown values in mathematical expressions and equations. The exercise involves identifying these unknowns. Let's break down the process:

• Step 1: Recognize what you're trying to find. Here, we need to find four consecutive integers.
• Step 2: Assign a variable to represent the smallest integer, which is x.
• Step 3: Identify the consecutive integers by adding 1, 2, and 3 to x. So, we get x, x+1, x+2, and x+3.

Identifying variables helps simplify complex problems. Instead of juggling multiple unknowns, we can manage the unknowns through one symbolic representation. This leads to easier formulation and solving of algebraic equations.

Solving Equations Efficiently

Solving equations involves finding the value of the variable that makes the equation true. Consider the equation derived in the exercise:
\[ x + (x+1) + (x+2) = (x+3) + 54 \]
Here’s how we solve it:

• Simplify: Combine like terms on both sides of the equation:
  \[ 3x + 3 = x + 57 \]
• Isolate the variable: Subtract x from both sides:
  \[ 3x + 3 - x = x + 57 - x \]
  This simplifies to:
  \[ 2x + 3 = 57 \]
• Continue simplifying: Subtract 3 from both sides to get:
  \[ 2x = 54 \]
• Solve for x: Divide both sides by 2:
  \[ x = 27 \]
• Find all values: Plug x back into the expressions for the consecutive integers to find:
  \[ 27, 28, 29, 30 \]

Double-checking solutions ensures accuracy. Verify that the sum of the first three integers is indeed 54 more than the fourth by computing:
\[ 27 + 28 + 29 = 84 \]
Then check:
\[ 84 - 30 = 54 \]
This confirms that the solution is correct.
Understanding these steps helps build a strong foundation in handling algebraic equations efficiently.