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Chapter 4: Problem 90

Solve each equation. $$(z+3)(z-3)=z(z-3)$$

Short Answer

Expert verified
The solution of the equation is \(z = -3\).

Step by step solution

01

Apply the Distributive Law

Start by applying the distributive law to the left side of the equation, getting \(z^2 - 3z + 3z - 9\) on the left side. The equation then reads \(z^2 - 9 = z^2 - 3z\).
02

Simplify the Equation

Now simplify the equation by bringing terms involving \(z\) on one side and constants on the other. The equation will then become \(0 = z^2 - z^2 + 3z + 9\). Simplify this further to \(0 = 3z + 9\).
03

Solve for z

Finally, solve for \(z\) by subtracting 9 from both sides and then dividing by 3. This gives \(z = -3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Law
The Distributive Law is a fundamental rule in algebra which helps you simplify expressions, especially when dealing with parentheses. You would use it when you want to multiply a single term by each term inside a parenthesis. The Distributive Law can be expressed as:
  • \( a(b + c) = ab + ac \)
In the given equation \((z+3)(z-3) = z(z-3)\), we apply the distributive law to both sides. For the left side, multiply \(z\) by both \(z-3\) and \(-3\), and \(+3\) by \(z-3\) and \(-3\). This breakdown results in
  • \(z^2 - 3z + 3z - 9\)
On the right side, only \(z\) is multiplies across, resulting in
  • \(z^2 - 3z\)
Understanding this step properly sets the stage for simplifying the equation next.
Simplifying Equations
Simplifying equations is key to making them easier to solve. It involves combining like terms and reducing equations to simpler forms. After applying the distributive law in the exercise, you encounter the equation:
\(z^2 - 9 = z^2 - 3z\).
You first identify like terms on both sides. Here, the \(z^2\) terms on both sides can cancel each other out. After cancellation, the equation remains as
  • \(0 = -3z + 9\)
We further simplify by moving all terms involving \(z\) to one side and constant terms to the other, resulting in
  • \(0 = 3z + 9\)
Simplifying helps clarify what needs to be done next in the process of solving the equation.
Solving Linear Equations
Solving linear equations involves isolating the variable on one side of the equation. After simplification in our example, we have
\(0 = 3z + 9\).
To find the value of \(z\), follow these steps:
  • Subtract 9 from both sides: \(0 - 9 = 3z\) simplifies to \(-9 = 3z\)
  • Divide both sides by 3: \(\frac{-9}{3} = z\) results in \(z = -3\)
This process solves the equation by making \(z\) alone on one side. It is crucial to perform the same operation on both sides to keep the equation balanced. By understanding these steps, you can tackle a wide range of similar algebraic equations with confidence.

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