Solve For $z$:$\[ \begin{array}{l} 3z - 2z = 17 \\ z = \square \end{array} \\]

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Solving Linear Equations: A Step-by-Step Guide to Finding the Value of z

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a simple linear equation to find the value of z. We will break down the problem into smaller steps, making it easy to understand and follow along.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form of ax + b = c, where a, b, and c are constants, and x is the variable. In our case, the equation is 3z - 2z = 17, where z is the variable we need to solve for.

Solving the Equation

To solve the equation, we need to isolate the variable z. We can start by combining like terms on the left-hand side of the equation.

Combining Like Terms

The equation is 3z - 2z = 17. We can combine the like terms 3z and -2z by adding their coefficients.

# Import necessary modules
import sympy as sp

# Define the variable
z = sp.symbols('z')

# Define the equation
equation = 3*z - 2*z - 17

# Combine like terms
simplified_equation = sp.simplify(equation)

The simplified equation is z = 17.

Solving for z

Now that we have the simplified equation, we can solve for z by isolating it on one side of the equation.

# Solve for z
solution = sp.solve(simplified_equation, z)

The solution is z = 17.

In this article, we solved a simple linear equation to find the value of z. We started by combining like terms on the left-hand side of the equation and then solved for z by isolating it on one side of the equation. The final solution is z = 17.

  • When solving linear equations, always start by combining like terms on the left-hand side of the equation.
  • Use the distributive property to expand expressions and simplify the equation.
  • Isolate the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.
  • Not combining like terms on the left-hand side of the equation.
  • Not isolating the variable on one side of the equation.
  • Not checking the solution for extraneous solutions.

Try solving the following linear equations:

  1. 2x + 5 = 11
  2. 3y - 2 = 7
  3. x/2 + 3 = 5

Solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve simple linear equations and apply the concepts to real-world problems. Remember to combine like terms, isolate the variable, and check for extraneous solutions. With practice, you will become proficient in solving linear equations and be able to tackle more complex problems.
Solving Linear Equations: A Q&A Guide

In our previous article, we discussed how to solve linear equations by combining like terms and isolating the variable. However, we know that practice makes perfect, and there's no better way to learn than by asking questions and getting answers. In this article, we'll address some common questions and concerns about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form of ax + b = c, where a, b, and c are constants, and x is the variable.

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, look for the following characteristics:

  • The highest power of the variable(s) is 1.
  • The equation can be written in the form of ax + b = c.
  • The equation does not contain any exponents or roots.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example:

  • Linear equation: 2x + 3 = 5
  • Quadratic equation: x^2 + 4x + 4 = 0

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, follow these steps:

  1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
  2. Simplify the equation by combining like terms.
  3. Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations when solving linear equations?

A: When solving linear equations, follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponents next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I check my solution for extraneous solutions?

A: To check your solution for extraneous solutions, plug the solution back into the original equation and verify that it is true. If the solution is not true, then it is an extraneous solution.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

  • Not combining like terms on the left-hand side of the equation.
  • Not isolating the variable on one side of the equation.
  • Not checking the solution for extraneous solutions.

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

  • Working through practice problems in a textbook or online resource.
  • Using online tools or apps to generate random linear equations.
  • Creating your own linear equations and solving them.

Solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article and practicing regularly, you can become proficient in solving linear equations and apply the concepts to real-world problems. Remember to combine like terms, isolate the variable, and check for extraneous solutions. With practice, you will become a master of solving linear equations!