Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-17,\) solve each equation or inequality. Each solution is an integer. $$ 9 a-7=29 $$

Step-by-Step Solution

Answer

The solution for the equation is \( a = 4 \).

Step 1: Rewrite the equation

Start with the given equation: \[ 9a - 7 = 29 \] Our goal is to solve for \( a \). The first step is to isolate the term containing \( a \).

Step 2: Add 7 to both sides

To isolate \( 9a \), add 7 to both sides of the equation: \[ 9a - 7 + 7 = 29 + 7 \] This simplifies to: \[ 9a = 36 \]

Step 3: Divide by 9

To solve for \( a \), divide both sides of the equation by 9: \[ \frac{9a}{9} = \frac{36}{9} \] This simplifies to: \[ a = 4 \]

Key Concepts

Integer Solutions

When solving algebraic equations, especially linear equations, one may frequently encounter integer solutions. An integer solution simply means that the value you find for the variable, like our variable "a" in this exercise, is a whole number. Whole numbers include

• positive numbers like 1, 2, 3,
• zero,
• and all negative numbers like -1, -2, -3.

These solutions are straightforward because you don't have to deal with fractions or decimals, and they often arise in problems that involve everyday scenarios or simple measurements. Integer solutions are central in fields like discrete mathematics and computer science, as these areas often deal with countable quantities.

Linear Equations

Linear equations are among the simplest forms of equations you'll encounter in algebra. They are called "linear" because their graph is a straight line. These equations are structured in the form \[ ax + b = c \]where:

• "a" and "b" are constants,
• "x" is the variable,
• and "c" is the resulting value when the equation is equalized.

In our exercise,

• "9a - 7 = 29", means "a" is our variable,
• 9 is the coefficient of "a", and both -7 and 29 are constants.

The beauty of linear equations lays in their simplicity - they only involve arithmetic operations: addition, subtraction, multiplication, or division. This makes them a fundamental tool in developing more complex problem-solving skills.

Step-by-Step Solutions

Breaking down solving processes into a series of clear and manageable steps is vital in understanding algebraic equations. Let's revisit how we approached the problem:1. **Rewriting the Equation**: We start with "9a - 7 = 29". It is essential to clearly state the problem you want to solve.2. **Isolating the Variable Term**: Next, we needed to isolate "9a" by adding 7 to both sides. This keeps the equation balanced as anything you do on one side, you must do on the other.3. **Solving for the Variable**: Finally, dividing by 9 gives "a = 4". This step required knowing basic arithmetic to simplify the equation.4. **Checking the Solution**: Even though it wasn't included originally, checking your answer ensures it's correct. You substitute back into the original equation: \( 9 \times 4 - 7 = 29 \). This confirms the solution.With step-by-step solutions, each part of the process focuses on one challenge at a time, making the problem less overwhelming. This structured approach not only aids in solving the given problem but also builds critical thinking and problem-solving skills for future tasks.