Find The Slope And \[$y\$\]-intercept Of Each Equation.11) \[$y + 3 = X\$\]12) \[$2y - 10 = -4x\$\]13) \[$-5 - Y = -3x\$\]14) \[$y = 5x\$\]15) \[$6 - 2y = -x\$\]16) \[$5y + 10 = -2x\$\]

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Introduction

In mathematics, the slope-intercept form of a linear equation is a powerful tool for understanding the relationship between two variables. The slope-intercept form is represented as y = mx + b, where m is the slope and b is the y-intercept. In this article, we will explore how to find the slope and y-intercept of various linear equations.

What is Slope and y-intercept?

  • Slope: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope is denoted by the letter m.
  • y-intercept: The y-intercept is the point at which the line intersects the y-axis. It is the value of y when x is equal to zero. The y-intercept is denoted by the letter b.

Finding Slope and y-intercept: Step-by-Step Guide

To find the slope and y-intercept of a linear equation, follow these steps:

  1. Rearrange the equation: Rearrange the equation to isolate y on one side of the equation.
  2. Identify the slope: Identify the coefficient of x as the slope (m).
  3. Identify the y-intercept: Identify the constant term as the y-intercept (b).

Example 1: Finding Slope and y-intercept of y + 3 = x

  • Rearrange the equation: Rearrange the equation to isolate y on one side of the equation: y = x - 3.
  • Identify the slope: Identify the coefficient of x as the slope (m): m = 1.
  • Identify the y-intercept: Identify the constant term as the y-intercept (b): b = -3.

Example 2: Finding Slope and y-intercept of 2y - 10 = -4x

  • Rearrange the equation: Rearrange the equation to isolate y on one side of the equation: 2y = -4x + 10, y = (-4/2)x + 5, y = -2x + 5.
  • Identify the slope: Identify the coefficient of x as the slope (m): m = -2.
  • Identify the y-intercept: Identify the constant term as the y-intercept (b): b = 5.

Example 3: Finding Slope and y-intercept of -5 - y = -3x

  • Rearrange the equation: Rearrange the equation to isolate y on one side of the equation: -y = -3x + 5, y = 3x - 5.
  • Identify the slope: Identify the coefficient of x as the slope (m): m = 3.
  • Identify the y-intercept: Identify the constant term as the y-intercept (b): b = -5.

Example 4: Finding Slope and y-intercept of y = 5x

  • Rearrange the equation: No need to rearrange the equation as it is already in slope-intercept form.
  • Identify the slope: Identify the coefficient of x as the slope (m): m = 5.
  • Identify the y-intercept: Identify the constant term as the y-intercept (b): b = 0.

Example 5: Finding Slope and y-intercept of 6 - 2y = -x

  • Rearrange the equation: Rearrange the equation to isolate y on one side of the equation: -2y = -x - 6, y = (1/2)x + 3.
  • Identify the slope: Identify the coefficient of x as the slope (m): m = 1/2.
  • Identify the y-intercept: Identify the constant term as the y-intercept (b): b = 3.

Example 6: Finding Slope and y-intercept of 5y + 10 = -2x

  • Rearrange the equation: Rearrange the equation to isolate y on one side of the equation: 5y = -2x - 10, y = (-2/5)x - 2.
  • Identify the slope: Identify the coefficient of x as the slope (m): m = -2/5.
  • Identify the y-intercept: Identify the constant term as the y-intercept (b): b = -2.

Conclusion

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, identify the coefficient of x as the slope (m). If the equation is in the form y = mx + b, the slope is m. If the equation is in the form y = -mx + b, the slope is -m.

Q: How do I find the y-intercept of a linear equation?

A: To find the y-intercept of a linear equation, identify the constant term as the y-intercept (b). If the equation is in the form y = mx + b, the y-intercept is b. If the equation is in the form y = mx - b, the y-intercept is -b.

Q: What is the difference between slope and y-intercept?

A: The slope of a line is a measure of how steep it is, while the y-intercept is the point at which the line intersects the y-axis.

Q: How do I find the slope and y-intercept of a linear equation with fractions?

A: To find the slope and y-intercept of a linear equation with fractions, follow the same steps as before. For example, if the equation is y = (1/2)x + 3, the slope is 1/2 and the y-intercept is 3.

Q: How do I find the slope and y-intercept of a linear equation with decimals?

A: To find the slope and y-intercept of a linear equation with decimals, follow the same steps as before. For example, if the equation is y = 0.5x + 3, the slope is 0.5 and the y-intercept is 3.

Q: Can I find the slope and y-intercept of a linear equation with negative numbers?

A: Yes, you can find the slope and y-intercept of a linear equation with negative numbers. For example, if the equation is y = -2x - 3, the slope is -2 and the y-intercept is -3.

Q: How do I graph a linear equation with a negative slope?

A: To graph a linear equation with a negative slope, follow these steps:

  1. Identify the slope: Identify the slope of the equation as negative.
  2. Identify the y-intercept: Identify the y-intercept of the equation.
  3. Plot the y-intercept: Plot the y-intercept on the graph.
  4. Plot a point: Plot a point on the graph that is one unit to the right of the y-intercept.
  5. Draw the line: Draw a line through the y-intercept and the point you plotted.

Q: How do I graph a linear equation with a positive slope?

A: To graph a linear equation with a positive slope, follow these steps:

  1. Identify the slope: Identify the slope of the equation as positive.
  2. Identify the y-intercept: Identify the y-intercept of the equation.
  3. Plot the y-intercept: Plot the y-intercept on the graph.
  4. Plot a point: Plot a point on the graph that is one unit to the right of the y-intercept.
  5. Draw the line: Draw a line through the y-intercept and the point you plotted.

Conclusion

In conclusion, finding the slope and y-intercept of a linear equation is a straightforward process that involves rearranging the equation to isolate y, identifying the coefficient of x as the slope, and identifying the constant term as the y-intercept. By following these steps, you can easily find the slope and y-intercept of various linear equations.