Solve The Following Linear Equation Using Equivalent Equations To Isolate The Variable. Express Your Answer As An Integer, As A Simplified Fraction, Or As A Decimal Number Rounded To Two Decimal Places.$[ -\frac{6}{7} T = \frac{2}{3}
Introduction
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 linear equations using equivalent equations to isolate the variable. We will use the given equation as an example and walk through the steps to find the solution.
Understanding the Problem
The given equation is:
Our goal is to isolate the variable t and express the solution as an integer, a simplified fraction, or a decimal number rounded to two decimal places.
Step 1: Multiply Both Sides by the Reciprocal of the Coefficient
To isolate the variable t, we need to get rid of the coefficient -6/7 that is being multiplied by t. We can do this by multiplying both sides of the equation by the reciprocal of the coefficient, which is 7/6.
-\frac{6}{7} t = \frac{2}{3}
\implies t = -\frac{6}{7} \times \frac{7}{6} \times \frac{2}{3}
Step 2: Simplify the Expression
Now, we can simplify the expression by canceling out the common factors.
t = -\frac{6}{7} \times \frac{7}{6} \times \frac{2}{3}
\implies t = -\frac{2}{3}
Step 3: Express the Solution as a Decimal Number
Finally, we can express the solution as a decimal number rounded to two decimal places.
t = -\frac{2}{3}
\implies t = -0.67
Conclusion
In this article, we solved the linear equation -6/7t = 2/3 using equivalent equations to isolate the variable t. We multiplied both sides of the equation by the reciprocal of the coefficient, simplified the expression, and finally expressed the solution as a decimal number rounded to two decimal places.
Tips and Tricks
- When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the calculations correctly.
- When multiplying both sides of an equation by a fraction, make sure to multiply both the numerator and the denominator by the fraction.
- When simplifying expressions, look for common factors to cancel out and simplify the expression.
Common Mistakes to Avoid
- Not following the order of operations (PEMDAS) when solving linear equations.
- Not multiplying both sides of an equation by the reciprocal of the coefficient.
- Not simplifying expressions by canceling out common factors.
Real-World Applications
Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. Some examples include:
- Modeling population growth and decline
- Calculating the trajectory of projectiles
- Determining the cost of goods and services
- Solving systems of linear equations to find the optimal solution.
Conclusion
Introduction
In our previous article, we discussed how to solve linear equations using equivalent equations to isolate the variable. In this article, we will provide a Q&A guide to help you better understand the concept and address any questions or concerns you may have.
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 ax + b = c, where a, b, and c are constants, and x is the variable.
Q: How do I solve a linear equation?
A: To solve a linear equation, you need to isolate the variable by getting rid of the constant term on the same side of the equation as the variable. 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 (PEMDAS)?
A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when solving an equation. The acronym PEMDAS stands for:
- 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 multiply both sides of an equation by a fraction?
A: To multiply both sides of an equation by a fraction, you need to multiply both the numerator and the denominator by the fraction. For example, if you have the equation x = 2, and you want to multiply both sides by 3/4, you would get:
x = 2
x = 2 × 3/4
x = 6/4
Q: How do I simplify an expression?
A: To simplify an expression, you need to look for common factors to cancel out. For example, if you have the expression 6/2 × 3/4, you can simplify it by canceling out the common factor of 2:
6/2 × 3/4
3 × 3/4
9/4
Q: What are some common mistakes to avoid when solving linear equations?
A: Some common mistakes to avoid when solving linear equations include:
- Not following the order of operations (PEMDAS)
- Not multiplying both sides of an equation by the reciprocal of the coefficient
- Not simplifying expressions by canceling out common factors
Q: How do I apply linear equations to real-world problems?
A: Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. Some examples include:
- Modeling population growth and decline
- Calculating the trajectory of projectiles
- Determining the cost of goods and services
- Solving systems of linear equations to find the optimal solution
Conclusion
S linear equations is a fundamental skill that has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can confidently solve linear equations and apply this skill to real-world problems. Remember to follow the order of operations, multiply both sides of the equation by the reciprocal of the coefficient, and simplify expressions by canceling out common factors. With practice and patience, you will become proficient in solving linear equations and be able to apply this skill to real-world problems.
Additional Resources
- Khan Academy: Linear Equations
- Mathway: Linear Equations
- Wolfram Alpha: Linear Equations
Practice Problems
- Solve the equation
2x + 3 = 5 - Solve the equation
x - 2 = 3 - Solve the equation
2x + 5 = 11
Answer Key
x = 1x = 5x = 3