Problem 83 - Intermediate Algebra

Question

Solve each equation. $$ a+18=5 a-3+a $$

Step-by-Step Solution

Answer

\( a = \frac{21}{5} \)

Step 1: Simplify the Equation

First, combine like terms on the right side of the equation. The equation is originally given as:\[ a + 18 = 5a - 3 + a \]Combine the \( a \) terms on the right side:\[ a + 18 = 6a - 3 \]

Step 2: Isolate Variables

Subtract \( a \) from both sides of the equation to get all \( a \) terms on one side:\[ 18 = 5a - 3 \]

Step 3: Isolate Constant Terms

Add 3 to both sides to eliminate the constant term on the right side:\[ 18 + 3 = 5a \]Simplify the left side:\[ 21 = 5a \]

Step 4: Solve for Variable

Divide by 5 on both sides to solve for \( a \):\[ a = \frac{21}{5} \]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of an unknown variable that makes the equation true. In the exercise we are tackling, the goal is to solve for the variable \(a\) in the equation, \(a + 18 = 5a - 3 + a\).

To solve an equation, we need to perform operations that simplify and reduce the equation step by step. Each operation should be done equally to both sides to maintain balance.

• Begin by identifying the terms on each side of the equation.
• Careful simplification is key to making the equation easier to solve.
• Consistently apply arithmetic operations like addition, subtraction, multiplication, or division.

Solving equations requires patience and practice, as these steps are foundational in intermediate algebra and more advanced math topics.

Combining Like Terms

Combining like terms is a critical step in simplifying an equation. It involves merging terms in an equation that have the same variable raised to the same power. In our exercise, the initially provided equation was:

\[a + 18 = 5a - 3 + a\]

On the right side, we notice both \(5a\) and \(a\) are like terms. By combining them, the equation simplifies to:

\[a + 18 = 6a - 3\]

When combining like terms, remember:

• Only terms with the same variable and exponent can be combined.
• Add or subtract their coefficients accordingly.
• This simplification allows for easier manipulation of the equation later.

Simplifying equations by combining like terms prepares the equation for further steps involving variable isolation or solving.

Isolation of Variables

The isolation of variables involves rearranging an equation such that the variable of interest is on one side of the equation, typically alone, making it straightforward to solve. During the exercise, this was achieved through several steps.

Starting with:
\[a + 18 = 6a - 3\]
To isolate \(a\), we first aimed to eliminate it from one side by subtracting \(a\) from both sides, leading to:
\[18 = 5a - 3\]
Then, we moved all constant terms to one side by adding 3 to both sides:
\[21 = 5a\]
Key Points to Isolating Variables:

• Always perform the same operation on both sides of the equation to keep it balanced.
• Work systematically to simplify the equation step by step.
• Each operation brings you closer to having the variable on its own.

By following these steps, you can successfully isolate variables in equations, paving the way for solving them.

Intermediate Algebra

Intermediate algebra is a branch of mathematics focusing on solving more complex equations involving variables. The concepts in this exercise are part of the foundational skills necessary for tackling intermediate algebra problems.

Here, we used a sequence of logical steps, starting from simplifying the equation to isolating the variable and solving for it. These form part of a bigger toolkit of algebraic techniques.

• Combining like terms helped us simplify the problem.
• Isolating variables allowed us to better navigate to the solution.
• Understanding these operations build up to more complex multi-step problems.

Intermediate algebra can be a stepping stone to advanced mathematics and applications in various fields such as science, engineering, finance, and economics.