Select The Correct Answer.What Is The Solution For $x$ In The Equation?$16x - 4 + 5x = -67$A. \$x = 3$[/tex\] B. $x = -\frac{1}{3}$ C. $x = -3$ D. $x = \frac{1}{3}$
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, $16x - 4 + 5x = -67$, and provide a step-by-step guide on how to arrive at the correct solution.
Understanding the Equation
The given equation is a linear equation in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants. In this case, $a = 16$, $b = -4$, and $c = -67$. The variable $x$ is the unknown quantity that we need to solve for.
Step 1: Combine Like Terms
The first step in solving the equation is to combine like terms. In this case, we have two terms with the variable $x$, which are $16x$ and $5x$. We can combine these terms by adding their coefficients, which gives us $21x$.
# Import necessary modules
import sympy as sp
# Define the variable
x = sp.symbols('x')
# Define the equation
equation = 16*x - 4 + 5*x + 67
# Combine like terms
simplified_equation = sp.simplify(equation)
print(simplified_equation)
Step 2: Isolate the Variable
After combining like terms, we get the simplified equation $21x - 4 + 67 = 0$. The next step is to isolate the variable $x$ by moving the constant term to the other side of the equation. We can do this by adding $4$ to both sides of the equation, which gives us $21x + 63 = 0$.
# Isolate the variable
isolated_equation = simplified_equation + 4
print(isolated_equation)
Step 3: Solve for x
The final step is to solve for $x$. We can do this by dividing both sides of the equation by $21$, which gives us $x = -\frac{63}{21}$.
# Solve for x
solution = sp.solve(isolated_equation, x)
print(solution)
Conclusion
In conclusion, the solution to the equation $16x - 4 + 5x = -67$ is $x = -\frac{63}{21}$, which simplifies to $x = -3$. This is the correct answer among the options provided.
Final Answer
The final answer is: C. $x = -3$
Discussion
This problem is a great example of how to solve linear equations by combining like terms and isolating the variable. The key concept here is to simplify the equation by combining like terms and then isolate the variable by moving the constant term to the other side of the equation. With practice and patience, students can master this skill and become proficient in solving linear equations.
Related Topics
- Solving quadratic equations
- Solving systems of linear equations
- Graphing linear equations
- Solving linear inequalities
Additional Resources
- Khan Academy: Solving Linear Equations
- Mathway: Solving Linear Equations
- Wolfram Alpha: Solving Linear Equations
Solving Linear Equations: A Q&A Guide =====================================
Introduction
Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will provide a Q&A guide on solving linear equations, covering common questions and topics.
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.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable by moving the constant term to the other side of the equation. You can do this 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: The order of operations when solving linear equations is:
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How do I combine like terms when solving linear equations?
A: To combine like terms, you need to add or subtract the coefficients of the same variable. For example, if you have the equation $2x + 3x = 5x$, you can combine the like terms by adding the coefficients, which gives you $5x$.
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, the equation $x + 2 = 3$ is a linear equation, while the equation $x^2 + 2x + 1 = 0$ is a quadratic equation.
Q: How do I graph a linear equation?
A: To graph a linear equation, you need to find two points on the line and plot them on a coordinate plane. You can then draw a line through the two points to represent the equation.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept.
Q: How do I find the slope of a linear equation?
A: To find the slope of a linear equation, you need to use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.
Q: What is the y-intercept of a linear equation?
A: The y-intercept of a linear equation is the point where the line intersects the y-axis. It is represented by the value of $b$ in the slope-intercept form of the equation.
Conclusion
In conclusion, solving linear equations is a crucial skill for students to master. By understanding the concepts and techniques covered in this Q&A guide, students can become proficient in solving linear equations and apply them to real-world problems.
Final Tips
- Practice, practice, practice: The more you practice solving linear equations, the more comfortable you will become with the concepts and techniques.
- Use online resources: There are many online resources available that can help you learn and practice solving linear equations, such as Khan Academy, Mathway, and Wolfram Alpha.
- Seek help when needed: Don't be afraid to ask for help if you are struggling with a particular concept or problem.