Isolating a variable like $z$ in an equation means solving for $z$ so that $z$ is on one side of the equation and everything else is on the other. Here’s a step-by-step guide to help you understand how to isolate $z$ in different types of equations.
Step-by-Step Process
1. Identify the Equation
First, identify the equation you are dealing with. For example, let’s start with a simple equation:
$3z + 5 = 20$
2. Simplify Both Sides
If there are any like terms on either side of the equation, combine them. In our example, the equation is already simplified.
3. Move Constants to the Other Side
Subtract or add constants to both sides to get $z$ alone. For our example:
$3z + 5 – 5 = 20 – 5$
This simplifies to:
$3z = 15$
4. Divide or Multiply to Solve for $z$
Divide or multiply both sides by the coefficient of $z$. In this case, divide both sides by 3:
$frac{3z}{3} = frac{15}{3}$
This simplifies to:
$z = 5$
Examples of Different Types of Equations
Linear Equations
For linear equations like $4z – 8 = 12$, follow the same steps:
- Add 8 to both sides: $4z – 8 + 8 = 12 + 8$
- Simplify: $4z = 20$
- Divide by 4: $z = frac{20}{4}$
- Solution: $z = 5$
Equations with Fractions
For equations with fractions like $frac{z}{2} + 3 = 7$:
- Subtract 3 from both sides: $frac{z}{2} + 3 – 3 = 7 – 3$
- Simplify: $frac{z}{2} = 4$
- Multiply both sides by 2: $z = 4 times 2$
- Solution: $z = 8$
Quadratic Equations
For quadratic equations like $z^2 – 4 = 0$:
- Add 4 to both sides: $z^2 – 4 + 4 = 0 + 4$
- Simplify: $z^2 = 4$
- Take the square root of both sides: $z = text{±} frac{text{√}4}{1}$
- Solution: $z = text{±} 2$
Conclusion
Isolating $z$ involves identifying the type of equation, simplifying both sides, moving constants, and then dividing or multiplying to get $z$ by itself. Whether you’re dealing with linear equations, equations with fractions, or even quadratic equations, the steps remain consistent. Practice these steps with different equations to become more comfortable with isolating variables.