Problem 41 - Algebra for College Students

Question

Solve each problem involving consecutive integers. Find three consecutive integers such that the sum of the first and twice the second is 17 more than twice the third.

Step-by-Step Solution

Answer

The three consecutive integers are 19, 20, and 21.

Step 1: Define Variables

Let the three consecutive integers be represented as follows: First integer: \( x \) Second integer: \( x + 1 \) Third integer: \( x + 2 \)

Step 2: Set Up the Equation

According to the problem, the sum of the first integer and twice the second integer is 17 more than twice the third integer. Therefore, set up the equation: \[ x + 2(x + 1) = 2(x + 2) + 17 \]

Step 3: Simplify the Equation

Simplify both sides of the equation: \[ x + 2x + 2 = 2x + 4 + 17 \] Combine like terms: \[ 3x + 2 = 2x + 21 \]

Step 4: Solve for \( x \)

To isolate \( x \), subtract \( 2x \) from both sides: \[ 3x + 2 - 2x = 2x + 21 - 2x \] This simplifies to: \[ x + 2 = 21 \] Subtract 2 from both sides to solve for \( x \): \[ x = 19 \]

Step 5: Find the Consecutive Integers

The first integer is \( 19 \). The second integer is \( 19 + 1 = 20 \). The third integer is \( 19 + 2 = 21 \).

Step 6: Verify the Solution

Check that the sum of the first integer and twice the second integer equals 17 more than twice the third integer: \[ 19 + 2(20) = 2(21) + 17 \] \[ 19 + 40 = 42 + 17 \] \[ 59 = 59 \] Both sides are equal, confirming the solution is correct.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. They often include variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. Solving algebraic equations is about finding the values of the variables that make the equation true.
In the problem about consecutive integers, we set up an algebraic equation based on the given conditions. Here, we represented the integers with variables (x, x+1, x+2) and formed an equation:
\[ x + 2(x + 1) = 2(x + 2) + 17 \]
This equation needs to be simplified step-by-step to solve for the unknown variable. Each step involves basic algebraic rules such as distributing multiplication across addition, combining like terms, and isolating the variable to one side of the equation.

Integer Sequences

Integer sequences are ordered lists of integers that follow a specific pattern or rule. Consecutive integers are a simple example of such sequences where each number is one more than the previous number. In mathematical terms, if the first integer is x, the next consecutive integers can be represented as x+1, x+2, and so on.
For example, if we start with 19, the consecutive integers would be 19, 20, and 21. Recognizing patterns in integer sequences can help set up algebraic equations correctly. By identifying the sequence in this problem, we can replace the integers by x, x+1, and x+2.
Understanding integer sequences helps break down complex problems into smaller, more manageable parts. This knowledge is particularly helpful in creating and solving equations as it ensures that we accurately represent the values we’re working with.

Linear Equations

Linear equations are equations of the first degree, which means they involve variables raised to the power of one. They typically take the form of ax + b = c, where a, b, and c are constants. Linear equations can be solved using various methods, including graphing, substitution, and elimination.
In the given problem, we derived and solved a linear equation:
\[ x + 2(x + 1) = 2(x + 2) + 17 \]
By simplifying it, we found that:
\[ 3x + 2 = 2x + 21 \]
Further simplification gives:
\[ x + 2 = 21 \]
Therefore, we isolate x by subtracting 2 from both sides: \[ x = 19 \]
Linear equations are foundational in algebra, providing a straightforward path to determine unknown values through systematic steps. Mastery of linear equations is crucial for tackling more advanced mathematical topics.