Select The Correct Answer From Each Drop-down Menu.The Given Equation Has Been Solved In The Table.$[ \begin{tabular}{|c|c|} \hline Step & Statement \ \hline 1 & − Y 2 − 6 = 15 -\frac{y}{2}-6=15 − 2 Y ​ − 6 = 15 \ \hline 2 & − Y 2 − 6 + 6 = 15 + 6 -\frac{y}{2}-6+6=15+6 − 2 Y ​ − 6 + 6 = 15 + 6 \ \hline 3 &

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Understanding the Basics of Linear Equations

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

The Given Equation

The given equation is −y2−6=15-\frac{y}{2}-6=15. This equation can be solved using basic algebraic operations. The goal is to isolate the variable y.

Step 1: Adding 6 to Both Sides

The first step in solving the equation is to add 6 to both sides of the equation. This will help us get rid of the negative term on the left-hand side.

−y2−6+6=15+6-\frac{y}{2}-6+6=15+6

This simplifies to:

−y2=21-\frac{y}{2}=21

Step 2: Multiplying Both Sides by -2

The next step is to multiply both sides of the equation by -2. This will help us get rid of the fraction on the left-hand side.

−y2×−2=21×−2-\frac{y}{2} \times -2 = 21 \times -2

This simplifies to:

y=−42y = -42

Conclusion

In conclusion, the correct solution to the given equation is y = -42. This can be verified by plugging the value of y back into the original equation.

Discussion Category: Mathematics

Mathematics is a vast and fascinating field that encompasses various branches, including algebra, geometry, calculus, and number theory. Linear equations are a fundamental concept in mathematics, and they have numerous applications in real-world problems.

Real-World Applications of Linear Equations

Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under the influence of forces.
  • 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.
  • Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Tips and Tricks for Solving Linear Equations

Here are some tips and tricks for solving linear equations:

  • Use inverse operations: Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division.
  • Use algebraic properties: Algebraic properties, such as the commutative and associative properties, can help simplify equations and make them easier to solve.
  • Check your work: Always check your work by plugging the solution back into the original equation.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving linear equations:

  • Not following the order of operations: The order of operations is a set of rules that dictate the order in which operations should be performed. Failing to follow the order of operations can lead to incorrect solutions.
  • Not checking your work: Failing to check your work can lead to incorrect solutions and a lack of confidence in your abilities.
  • Not using inverse operations: Failing to use inverse operations can make it difficult to solve equations and may lead to incorrect solutions.

Conclusion

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that 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 performing inverse operations. This may involve adding or subtracting the same value to both sides of the equation, or multiplying or dividing both sides by the same non-zero value.

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
  • Not checking your work
  • Not using inverse operations
  • Not simplifying the equation before solving it

Q: How do I check my work when solving a linear equation?

A: To check your work when solving a linear equation, plug the solution back into the original equation and verify that it is true. If the solution is not true, then you need to re-solve the equation.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under the influence of forces.
  • 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.
  • Computer Science: Linear equations are used in computer graphics, game development, and machine learning.

Q: How do I simplify a linear equation before solving it?

A: To simplify a linear equation before solving it, combine like terms and eliminate any fractions by multiplying both sides of the equation by the denominator.

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.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, use the method of substitution or elimination to find the values of the variables.

Q: What are some tips for solving linear equations?

A: Some tips for solving linear equations include:

  • Use inverse operations to isolate the variable
  • Simplify the equation before solving it
  • Check your work by plugging the solution back into the original equation
  • Use algebraic properties, such as the commutative and associative properties, to simplify the equation

Q: How do I graph a linear equation?

A: To graph a linear equation, use the slope-intercept form of the equation, which is y = mx + b, where m is the slope and b is the y-intercept.

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 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, use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) 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 denoted by the letter b in the slope-intercept form of the equation, y = mx + b.