Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-17,\) solve each equation or inequality. Each solution is an integer. $$ 7 x+18=39 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 3 \).

Step 1: Understand the Equation

Identify the equation given in the problem, which is: \( 7x + 18 = 39 \). Our goal is to solve for \( x \).

Step 2: Isolate the Variable Term

To solve for \( x \), we need to isolate the variable term. Start by removing the constant term on the left side of the equation. Subtract 18 from both sides:\[7x + 18 - 18 = 39 - 18\]This simplifies to:\[7x = 21\]

Step 3: Solve for the Variable

With \( 7x = 21 \), divide both sides of the equation by 7 to solve for \( x \):\[\frac{7x}{7} = \frac{21}{7}\]This simplifies to:\[x = 3\]

Step 4: Verify the Solution

Go back to the original equation and substitute \( x = 3 \) to check the solution:\[7(3) + 18 = 39\]Calculate the left side:\[21 + 18 = 39\]Since both sides equal 39, \( x = 3 \) is verified as the correct solution.

Key Concepts

Solving Equations

Solving an equation means finding a value for the variable that makes the equation true. Imagine it as a balance scale, where both sides must always stay equal. To solve the equation \(7x + 18 = 39\), we need to find the value of \(x\) that makes both sides of the equation equal.

The step-by-step approach begins with identifying the equation precisely. Here, we see that after clarifying the equation as given, our task is to isolate the variable \(x\). This means we'll rearrange the equation until \(x\) is on one side by itself.
A major part of solving involves reversing operations. Since \(18\) is added to \(7x\), we do the opposite by subtracting \(18\) from both sides to maintain balance.

• This results in \(7x = 21\), a much simpler equation where \(x\) can be isolated through division.
• Finally, by dividing both sides by \(7\), we solve for \(x\), which equals \(3\).

Solving equations involves understanding these inverse operations and keeping the equation's balance intact.

Integer Solutions

In the process of solving equations, sometimes you will find that solutions are integers. An integer is any whole number, negative, zero, or positive, without fractions or decimals.

In our example, once we have simplified the equation to \(7x = 21\), the next step of dividing both sides by \(7\) results in \(x = 3\), a straightforward integer solution.

• Integer solutions offer the simplest form of answers, as they are easy to understand and represent.
• They signify points where the equation crosses whole number values, which can be particularly useful in many real-world applications.

In math, finding an integer solution implies that the answer is precise and doesn't need further rounding or approximation. This is beneficial in contexts where exactness is crucial.

Equation Verification

Verification is an essential step in any algebra problem, confirming that the solution you found is correct.

To verify, substitute the found value back into the original equation. For our solution where \(x = 3\), we place \(3\) back into the original equation \(7x + 18 = 39\).

• Substituting gives \(7(3) + 18 = 39\).
• Calculating the left side, \(21 + 18\) indeed equals \(39\), confirming the equations's balance and correctness.

Verification ensures that no mistakes were made during solving. It gives you confidence and clarity that \(x = 3\) is not just a plausible but a mathematically sound solution. Always remember, verifying is a great practice for accuracy and learning.