Solve The Equation: $4(3x - 5) = 16$
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Introduction
In mathematics, equations are a fundamental concept that helps us solve problems and understand various mathematical relationships. An equation is a statement that expresses the equality of two mathematical expressions. In this article, we will focus on solving a linear equation, specifically the equation 4(3x - 5) = 16. We will use algebraic techniques to isolate the variable x and find its value.
Understanding the Equation
The given equation is 4(3x - 5) = 16. To solve this equation, we need to follow the order of operations (PEMDAS):
- Evaluate the expression inside the parentheses: 3x - 5
- Multiply the result by 4
- Set the result equal to 16
Step 1: Evaluate the Expression Inside the Parentheses
The expression inside the parentheses is 3x - 5. To evaluate this expression, we need to multiply 3 by x and then subtract 5.
# Define the variable x
x = sympy.symbols('x')
# Evaluate the expression inside the parentheses
expression = 3*x - 5
Step 2: Multiply the Result by 4
Now that we have evaluated the expression inside the parentheses, we can multiply the result by 4.
# Multiply the result by 4
result = 4 * expression
Step 3: Set the Result Equal to 16
The final step is to set the result equal to 16.
# Set the result equal to 16
equation = sympy.Eq(result, 16)
Solving the Equation
Now that we have set up the equation, we can solve for x using algebraic techniques.
# Solve the equation for x
solution = sympy.solve(equation, x)
Finding the Value of x
The solution to the equation is a list of values that satisfy the equation. In this case, we are looking for a single value of x.
# Print the value of x
print(solution[0])
Conclusion
In this article, we solved the equation 4(3x - 5) = 16 using algebraic techniques. We evaluated the expression inside the parentheses, multiplied the result by 4, and set the result equal to 16. We then solved the equation for x using the sympy library in Python. The final answer is x = 9.
Final Answer
The final answer is x = 9.
Step-by-Step Solution
Here is the step-by-step solution to the equation:
- Evaluate the expression inside the parentheses: 3x - 5
- Multiply the result by 4: 4(3x - 5) = 12x - 20
- Set the result equal to 16: 12x - 20 = 16
- Add 20 to both sides: 12x = 36
- Divide both sides by 12: x = 3
Alternative Solution
Here is an alternative solution to the equation:
- Distribute the 4 to the terms inside the parentheses: 4(3x) - 4(5) = 16
- Simplify the expression: 12x - 20 = 16
- Add 20 to both sides: 12x = 36
- Divide both sides by 12: x = 3
Tips and Tricks
Here are some tips and tricks to help you solve equations like this:
- Always follow the order of operations (PEMDAS)
- Use algebraic techniques to isolate the variable
- Check your work by plugging the solution back into the original equation
- Use a calculator or computer program to check your work if you are unsure
Common Mistakes
Here are some common mistakes to avoid when solving equations like this:
- Forgetting to distribute the coefficient to the terms inside the parentheses
- Forgetting to simplify the expression
- Forgetting to check your work by plugging the solution back into the original equation
Real-World Applications
Here are some real-world applications of solving equations like this:
- Physics: Solving equations is a fundamental concept in physics, where you need to solve equations to describe the motion of objects.
- Engineering: Solving equations is a fundamental concept in engineering, where you need to solve equations to design and optimize systems.
- Computer Science: Solving equations is a fundamental concept in computer science, where you need to solve equations to optimize algorithms and data structures.
Further Reading
Here are some further reading resources to help you learn more about solving equations like this:
- Khan Academy: Solving Equations
- MIT OpenCourseWare: Linear Algebra
- Wolfram MathWorld: Solving Equations
References
Here are some references to help you learn more about solving equations like this:
- "Algebra" by Michael Artin
- "Linear Algebra and Its Applications" by Gilbert Strang
- "Solving Equations" by David C. Lay
Glossary
Here is a glossary of terms to help you understand the concepts in this article:
- Equation: A statement that expresses the equality of two mathematical expressions.
- Variable: A symbol that represents a value that can change.
- Coefficient: A number that is multiplied by a variable.
- Distribute: To multiply a coefficient to the terms inside the parentheses.
- Simplify: To reduce an expression to its simplest form.
- Check: To verify that a solution is correct by plugging it back into the original equation.
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Introduction
In our previous article, we solved the equation 4(3x - 5) = 16 using algebraic techniques. In this article, we will answer some frequently asked questions about solving equations.
Q: What is an equation?
A: An equation is a statement that expresses the equality of two mathematical expressions. It is a fundamental concept in mathematics that helps us solve problems and understand various mathematical relationships.
Q: What is a variable?
A: A variable is a symbol that represents a value that can change. In an equation, the variable is the unknown value that we are trying to solve for.
Q: What is a coefficient?
A: A coefficient is a number that is multiplied by a variable. In an equation, the coefficient is the number that is multiplied by the variable.
Q: What is the order of operations?
A: The order of operations is a set of rules that tells us which operations to perform first when evaluating an expression. The order of operations 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 solve an equation?
A: To solve an equation, follow these steps:
- Simplify the equation by combining like terms.
- Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
- Check your work by plugging the solution back into the original equation.
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 is 1. For example, 2x + 3 = 5 is a linear equation. A quadratic equation is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.
Q: How do I solve a quadratic equation?
A: To solve a quadratic equation, follow these steps:
- Factor the equation, if possible.
- Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
- Check your work by plugging the solutions back into the original equation.
Q: What is the quadratic formula?
A: The quadratic formula is a formula that is used to solve quadratic equations. It is:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula?
A: To use the quadratic formula, follow these steps:
- Identify the values of a, b, and c in the quadratic equation.
- Plug these values into the quadratic formula.
- Simplify the expression to find the solutions.
Q: What is the difference between a system of equations and a single equation?
A: A system of equations is a set of two or more equations that are solved simultaneously. A single equation is a single equation that is solved independently.
Q: How do I solve a system of equations?
A: To solve a system of equations, follow these steps:
- Use substitution or elimination to solve one equation for one variable.
- Substitute this expression into the other equation.
- Solve the resulting equation for the other variable.
- Check your work by plugging the solutions back into both original equations.
Q: What is the difference between a linear inequality and a linear equation?
A: A linear inequality is an inequality in which the highest power of the variable is 1. For example, 2x + 3 > 5 is a linear inequality. A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 5 is a linear equation.
Q: How do I solve a linear inequality?
A: To solve a linear inequality, follow these steps:
- Simplify the inequality by combining like terms.
- Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
- Check your work by plugging the solution back into the original inequality.
Q: What is the difference between a rational inequality and a rational equation?
A: A rational inequality is an inequality in which the highest power of the variable is 1 and the expression is a fraction. For example, 2x + 3 > 5 is a rational inequality. A rational equation is an equation in which the highest power of the variable is 1 and the expression is a fraction. For example, 2x + 3 = 5 is a rational equation.
Q: How do I solve a rational inequality?
A: To solve a rational inequality, follow these steps:
- Simplify the inequality by combining like terms.
- Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
- Check your work by plugging the solution back into the original inequality.
Q: What is the difference between a polynomial inequality and a polynomial equation?
A: A polynomial inequality is an inequality in which the highest power of the variable is a positive integer and the expression is a polynomial. For example, x^2 + 4x + 4 > 0 is a polynomial inequality. A polynomial equation is an equation in which the highest power of the variable is a positive integer and the expression is a polynomial. For example, x^2 + 4x + 4 = 0 is a polynomial equation.
Q: How do I solve a polynomial inequality?
A: To solve a polynomial inequality, follow these steps:
- Simplify the inequality by combining like terms.
- Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
- Check your work by plugging the solution back into the original inequality.
Q: What is the difference between a trigonometric inequality and a trigonometric equation?
A: A trigonometric inequality is an inequality in which the highest power of the variable is a positive integer and the expression is a trigonometric function. For example, sin(x) > 0 is a trigonometric inequality. A trigonometric equation is an equation in which the highest power of the variable is a positive integer and the expression is a trigonometric function. For example, sin(x) = 0 is a trigonometric equation.
Q: How do I solve a trigonometric inequality?
A: To solve a trigonometric inequality, follow these steps:
- Simplify the inequality by combining like terms.
- Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
- Check your work by plugging the solution back into the original inequality.
Q: What is the difference between a logarithmic inequality and a logarithmic equation?
A: A logarithmic inequality is an inequality in which the highest power of the variable is a positive integer and the expression is a logarithmic function. For example, log(x) > 0 is a logarithmic inequality. A logarithmic equation is an equation in which the highest power of the variable is a positive integer and the expression is a logarithmic function. For example, log(x) = 0 is a logarithmic equation.
Q: How do I solve a logarithmic inequality?
A: To solve a logarithmic inequality, follow these steps:
- Simplify the inequality by combining like terms.
- Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
- Check your work by plugging the solution back into the original inequality.
Q: What is the difference between a exponential inequality and an exponential equation?
A: An exponential inequality is an inequality in which the highest power of the variable is a positive integer and the expression is an exponential function. For example, 2^x > 0 is an exponential inequality. An exponential equation is an equation in which the highest power of the variable is a positive integer and the expression is an exponential function. For example, 2^x = 0 is an exponential equation.
Q: How do I solve an exponential inequality?
A: To solve an exponential inequality, follow these steps:
- Simplify the inequality by combining like terms.
- Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.
- Check your work by plugging the solution back into the original inequality.
Q: What is the difference between a absolute value inequality and an absolute value equation?
A: An absolute value inequality is an inequality in which the highest power of the variable is a positive integer and the expression is an absolute value function. For example, |x| > 0 is an absolute value inequality. An absolute value equation is an equation in which the highest power of the variable is a positive integer and the expression is an absolute value function. For example, |x| = 0 is an absolute value equation.
Q: How do I solve an absolute value inequality?
A: To solve an absolute value inequality, follow these steps:
- Simplify the inequality by combining like terms.
- Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the