For $y = -\frac{1}{2}x + 4$, Calculate The Following $y$ Values:A. $(-2, \square$\]B. $(12, \square$\]

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Introduction

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on solving linear equations in the form of y = mx + b, where m is the slope and b is the y-intercept. We will use the given equation y = -1/2x + 4 as an example to calculate the y-values for specific x-values.

Understanding the Equation

The given equation is y = -1/2x + 4, where m = -1/2 and b = 4. The slope (m) represents the rate of change of the line, and the y-intercept (b) represents the point where the line intersects the y-axis.

Calculating y-Values

To calculate the y-values for specific x-values, we can substitute the x-values into the equation and solve for y.

A. Calculating y for x = -2

To calculate the y-value for x = -2, we can substitute x = -2 into the equation:

y = -1/2(-2) + 4

First, we multiply -1/2 by -2:

-1/2(-2) = 1

Now, we add 4 to the result:

y = 1 + 4 y = 5

Therefore, the y-value for x = -2 is 5.

B. Calculating y for x = 12

To calculate the y-value for x = 12, we can substitute x = 12 into the equation:

y = -1/2(12) + 4

First, we multiply -1/2 by 12:

-1/2(12) = -6

Now, we add 4 to the result:

y = -6 + 4 y = -2

Therefore, the y-value for x = 12 is -2.

Conclusion

In this article, we have learned how to solve linear equations in the form of y = mx + b. We have used the given equation y = -1/2x + 4 as an example to calculate the y-values for specific x-values. By following the steps outlined in this article, you can easily solve linear equations and calculate y-values for various x-values.

Tips and Tricks

  • Make sure to substitute the x-values into the equation correctly.
  • Follow the order of operations (PEMDAS) when solving the equation.
  • Use a calculator or a graphing tool to visualize the line and check your results.

Real-World Applications

Linear equations have numerous 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.

Common Mistakes

  • Failing to substitute the x-values into the equation correctly.
  • Not following the order of operations (PEMDAS) when solving the equation.
  • Not checking the results using a calculator or a graphing tool.

A. Solving Linear Equations with One Variable

To solve a linear equation with one variable, we can use the following steps:

  1. Simplify the equation by combining like terms.
  2. Isolate the variable by adding or subtracting the same value to both sides of the equation.
  3. Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

B. Solving Linear Equations with Two Variables

To solve a linear equation with two variables, we can use the following steps:

  1. Simplify the equation by combining like terms.
  2. Isolate one of the variables by adding or subtracting the same value to both sides of the equation.
  3. Solve for the other variable by substituting the value of the isolated variable into the equation.

C. Solving Linear Equations with No Solution

To solve a linear equation with no solution, we can use the following steps:

  1. Simplify the equation by combining like terms.
  2. Check if the equation is inconsistent (i.e., the two sides of the equation are not equal).
  3. If the equation is inconsistent, then there is no solution.

Conclusion

Introduction

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will provide a Q&A guide to help you understand and solve linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (x) is 1. It can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the slope (m) in a linear equation?

A: The slope (m) represents the rate of change of the line. It can be positive, negative, or zero.

Q: What is the y-intercept (b) in a linear equation?

A: The y-intercept (b) represents the point where the line intersects the y-axis.

Q: How do I solve a linear equation?

A: To solve a linear equation, you can use the following steps:

  1. Simplify the equation by combining like terms.
  2. Isolate the variable by adding or subtracting the same value to both sides of the equation.
  3. Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

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 (x) is 1, while a quadratic equation is an equation in which the highest power of the variable (x) is 2.

Q: Can a linear equation have no solution?

A: Yes, a linear equation can have no solution if the two sides of the equation are not equal.

Q: How do I determine if a linear equation has no solution?

A: To determine if a linear equation has no solution, you can check if the equation is inconsistent (i.e., the two sides of the equation are not equal).

Q: What is the significance of linear equations in real-world applications?

A: Linear equations have numerous real-world applications, including physics, engineering, and economics.

Q: Can I use a calculator or a graphing tool to solve linear equations?

A: Yes, you can use a calculator or a graphing tool to solve linear equations and visualize the line.

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

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

  • Failing to substitute the x-values into the equation correctly.
  • Not following the order of operations (PEMDAS) when solving the equation.
  • Not checking the results using a calculator or a graphing tool.

Conclusion

In this article, we have provided a Q&A guide to help you understand and solve linear equations. By following the steps outlined in this article, you can easily solve linear equations and calculate y-values for various x-values.

Tips and Tricks

  • Make sure to substitute the x-values into the equation correctly.
  • Follow the order of operations (PEMDAS) when solving the equation.
  • Use a calculator or a graphing tool to visualize the line and check your results.

Real-World Applications

Linear equations have numerous 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.

Common Mistakes

  • Failing to substitute the x-values into the equation correctly.
  • Not following the order of operations (PEMDAS) when solving the equation.
  • Not checking the results using a calculator or a graphing tool.