Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation (if possible). $$4(z-3)+3 z=1+8 z$$

Step-by-Step Solution

Answer

The solution to the equation is \(z = -13\).

Step 1: Simplify the Equation

Start by simplifying both sides of the equation. In the left side of the equation, distribute 4 to both (z-3) and 3z resulting to \(4z - 12 + 3z\). Hence, the equation becomes: \(4z - 12 + 3z = 1+8z\).

Step 2: Combine Like Terms

Next, combine like terms. On the left, bring together the z-terms. \(4z + 3z\) gives \(7z - 12\). In the right side of the equation, the equation is already simplified. The equation now is: \(7z - 12 = 1 + 8z\).

Step 3: Isolate the Variable

The final step involves rearranging the equation once again in a manner that isolates the variable z. Subtract \(8z\) from both sides and add \(12\) on both sides to achieve this. Doing this yields the equation: \(7z - 8z = 1 + 12\). Solving this gives: \(-z = 13\). Finally, divide by \(-1\) on both sides of the equation to solve for z hence, \(z = -13\).

Key Concepts

Combining Like Terms

In the art of solving linear equations, combining like terms is a crucial step. Let’s break it down. Like terms are terms that have the same variables raised to the same power. In simpler terms, they look the same except for their coefficients. For instance, in our example, the terms \(4z\) and \(3z\) are like terms because both terms contain the variable \(z\) raised to the power of one.

When we combine like terms, we perform arithmetic operations on the coefficients while keeping the variable part the same. In the equation \(7z - 12 + 3z = 1 + 8z\), we combine \(4z\) and \(3z\) to get \(7z\).

• Identify like terms on both sides of the equation.
• Add or subtract the coefficients of the like terms together.
• Remember, combining like terms simplifies the equation and makes it easier to solve.

Through this process, equations become simpler and more manageable, allowing us to proceed to the next step: applying further operations as needed, like using the distributive property or isolating variables.

Distributive Property

The distributive property is an essential tool in algebra that helps in simplifying expressions and solving equations. It involves multiplying a single term by each term inside a set of parentheses.

In the example equation \(4(z-3) + 3z = 1 + 8z\), we apply the distributive property to the term \(4(z-3)\). Here’s how it works:

• Multiply \(4\) by both \(z\) and \(-3\).
• So, you compute: \(4 \times z\) gives \(4z\), and \(4 \times -3\) gives \(-12\).
• This results in \(4z - 12\).

Using the distributive property helps break down complex expressions, distributing and simplifying terms, hence aiding the transition to subsequent steps like combining like terms or isolating variables.

Keep in mind that the distributive property is a powerful technique that can be used whenever we see multiplication distributed over addition or subtraction in algebraic expressions.

Isolating Variable

Isolating the variable is a fundamental step to solving an equation. It means getting the variable we wish to solve for, usually \(z\), alone on one side of the equation. This is essential for determining its value.

In our specific equation, \(7z - 12 = 1 + 8z\):

• First, we subtract \(8z\) from both sides to bring the \(z\) terms together. This leaves us with \(7z - 8z\).
• Then, on simplifying, we get \(-z = 1 + 12\).
• We add \(12\) to \(1\) on the right, making it \(13\).
• Next, divide both sides by \(-1\) to isolate \(z\), resulting in \(z = -13\).

Isolating \(z\) involves carefully using addition and subtraction while respecting both sides of the equation. The goal is to simplify until the unknown variable stands alone, and calculations can directly reveal its value. By mastering this skill, you can solve linear equations with clarity and efficiency.