Q45. - Algebra 1

Question

Write an equation and solve each problem.

Find the sum of three consecutive odd integers if the sum of the first two integers is equal to twenty-four less than four times the third integer.

Step-by-Step Solution

Answer

The three numbers are $5,7,9$.

Step 1. Assign variables for the numbers.

A variable is a letter or a literal used to signify an unknown quantity.

Let letter $x$ represent the first number then next two odd consecutive numbers will be $x+2$ and $x+4$. 

Step 2. Form an equation for the sum of numbers.

It is given that the sum of the first two integers is equal to twenty-four less than four times the third integer thus, we have:

$x+\left(x+2\right)=4\left(x+4\right)-24$

Step 3. Simplify the equation.

We obtained the equation $x+\left(x+2\right)=4\left(x+4\right)-24$, to simplify it, we open the brackets as shown:

$\begin{array}{l}x+\left(x+2\right)=4\left(x+4\right)-24\\ x+x+2=4x+16-24\end{array}$

Like terms are the terms having same variables and is raised to the same index. Coefficient is the appended number in front of the variable; if no number is shown then we assume it to be 1.

Unlike terms are the terms not having same variables and or are raised to the different index.

For example 4 and $5x$ are unlike terms.

We can add like terms by adding their coefficients, we cannot add unlike terms.

Now, we are having the expression $x+x+2=4x+16-24$, combining like terms we get:

$\begin{array}{l}x+x+2=4x+16-24\\ 2x+2=4x-8\end{array}$

Step 4. Solve the equation.

Now, we are having the expression $2x+2=4x-8$, adding $8$ both sides we get:

$\begin{array}{l}2x+2+8=4x-8+8\\ x+10=4x\end{array}$

Now we are having the expression $2x+10=4x$, adding $-2x$ both sides we get:

$\begin{array}{l}2x+10-2x=4x-2x\\ 10=2x\end{array}$

Dividing both sides by 2 we get:

$\begin{array}{l}\frac{10}{2}=\frac{2x}{2}\\ 5=x\\ x=5\end{array}$

Thus, the first number is $5$.

Step 5. Use the value of the first number to get the other numbers.

We know that the first number is $5$, thus the other consecutive numbers must be $7$ and $9$.