What Is The Slope Of The Line With The Equation $y - 3 = -\frac{1}{2}(x - 2$\]?

What is the Slope of the Line with the Equation $y - 3 = -\frac{1}{2}(x - 2)$?

The slope of a line is a fundamental concept in mathematics, particularly in algebra and geometry. It represents the rate of change of the line's y-coordinate with respect to its x-coordinate. In this article, we will explore how to find the slope of a line given its equation in the form of $y - 3 = -\frac{1}{2}(x - 2)$.

The given equation is in the form of a linear equation, which is $y - 3 = -\frac{1}{2}(x - 2)$. To find the slope of this line, we need to rewrite the equation in the slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Rewriting the Equation

To rewrite the equation, we need to isolate the y-variable on one side of the equation. We can do this by adding $\frac{1}{2}(x - 2)$ to both sides of the equation.

y - 3 = -\frac{1}{2}(x - 2)
y - 3 + \frac{1}{2}(x - 2) = -\frac{1}{2}(x - 2) + \frac{1}{2}(x - 2)
y - 3 + \frac{1}{2}x - 1 = 0
y = \frac{1}{2}x - 4

Now that we have rewritten the equation in the slope-intercept form, we can easily identify the slope. In the equation $y = \frac{1}{2}x - 4$, the slope is the coefficient of the x-variable, which is $\frac{1}{2}$.

The slope of the line represents the rate of change of the y-coordinate with respect to the x-coordinate. In this case, the slope is $\frac{1}{2}$, which means that for every unit increase in the x-coordinate, the y-coordinate increases by $\frac{1}{2}$ unit.

The concept of slope has numerous real-world applications, including:

• Physics: The slope of a line can represent the rate of change of velocity or acceleration.
• Economics: The slope of a line can represent the rate of change of price or quantity demanded.
• Computer Science: The slope of a line can represent the rate of change of a function or algorithm.

In conclusion, the slope of the line with the equation $y - 3 = -\frac{1}{2}(x - 2)$ is $\frac{1}{2}$. This represents the rate of change of the y-coordinate with respect to the x-coordinate. The concept of slope has numerous real-world applications and is a fundamental concept in mathematics.

For further reading on the concept of slope, we recommend the following resources:

• Algebra textbooks: Many algebra textbooks cover the concept of slope in detail.
• Online resources: Websites such as Khan Academy and Mathway offer interactive lessons and exercises on the concept of slope.
• Mathematical software: Software such as Mathematica and Maple can be used to visualize and explore the concept of slope.

Q: What is the slope of a horizontal line? A: The slope of a horizontal line is 0.

Q: What is the slope of a vertical line? A: The slope of a vertical line is undefined.

Q: How do I find the slope of a line given its equation? A: To find the slope of a line given its equation, rewrite the equation in the slope-intercept form and identify the coefficient of the x-variable.

• Slope: The rate of change of the y-coordinate with respect to the x-coordinate.
• Linear equation: An equation in which the highest power of the variable is 1.
• Slope-intercept form: A form of a linear equation in which the y-variable is isolated on one side of the equation.
  Frequently Asked Questions: Slope of a Line =====================================================

Q: What is the slope of a horizontal line?

A: The slope of a horizontal line is 0. This is because the y-coordinate does not change with respect to the x-coordinate.

Q: What is the slope of a vertical line?

A: The slope of a vertical line is undefined. This is because the x-coordinate does not change with respect to the y-coordinate.

Q: How do I find the slope of a line given its equation?

A: To find the slope of a line given its equation, rewrite the equation in the slope-intercept form and identify the coefficient of the x-variable.

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 determine the slope of a line from its graph?

A: To determine the slope of a line from its graph, choose two points on the line and calculate the ratio of the vertical change to the horizontal change.

Q: What is the difference between the slope and the rate of change?

A: The slope and the rate of change are related but distinct concepts. The slope represents the rate of change of the y-coordinate with respect to the x-coordinate, while the rate of change represents the change in the y-coordinate over a given interval.

Q: Can the slope of a line be negative?

A: Yes, the slope of a line can be negative. This means that the y-coordinate decreases as the x-coordinate increases.

Q: Can the slope of a line be zero?

A: Yes, the slope of a line can be zero. This means that the y-coordinate does not change with respect to the x-coordinate.

Q: Can the slope of a line be undefined?

A: Yes, the slope of a line can be undefined. This means that the x-coordinate does not change with respect to the y-coordinate.

Q: How do I use the slope to predict the future behavior of a line?

A: To use the slope to predict the future behavior of a line, use the equation y = mx + b, where m is the slope and b is the y-intercept. Plug in a value for x and solve for y to find the corresponding value of y.

Q: Can the slope of a line be used to determine the equation of the line?

A: Yes, the slope of a line can be used to determine the equation of the line. Use the equation y = mx + b, where m is the slope and b is the y-intercept, and plug in the values of m and b to find the equation of the line.

Q: How do I use the slope to determine the equation of a line given two points?

A: To use the slope to determine the equation of a line given two points, choose two points on the line and calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Then, use the equation y = mx + b and plug in the values of m and the coordinates of one of the points to find the equation of the line.

Q: Can the slope of a line be used to determine the equation of a line given a point and a slope?

A: Yes, the slope of a line can be used to determine the equation of a line given a point and a slope. Use the equation y = mx + b, where m is the slope and b is the y-intercept, and plug in the values of m and the coordinates of the point to find the equation of the line.

Q: How do I use the slope to determine the equation of a line given a point and a slope and a second point?

A: To use the slope to determine the equation of a line given a point and a slope and a second point, choose two points on the line and calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Then, use the equation y = mx + b and plug in the values of m and the coordinates of one of the points to find the equation of the line.

Q: Can the slope of a line be used to determine the equation of a line given a point and a slope and a second point and a third point?

A: Yes, the slope of a line can be used to determine the equation of a line given a point and a slope and a second point and a third point. Choose three points on the line and calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Then, use the equation y = mx + b and plug in the values of m and the coordinates of one of the points to find the equation of the line.

Q: How do I use the slope to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point?

A: To use the slope to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point, choose four points on the line and calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Then, use the equation y = mx + b and plug in the values of m and the coordinates of one of the points to find the equation of the line.

Q: Can the slope of a line be used to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point?

A: Yes, the slope of a line can be used to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point. Choose five points on the line and calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Then, use the equation y = mx + b and plug in the values of m and the coordinates of one of the points to find the equation of the line.

Q: How do I use the slope to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point and a sixth point?

A: To use the slope to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point and a sixth point, choose six points on the line and calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Then, use the equation y = mx + b and plug in the values of m and the coordinates of one of the points to find the equation of the line.

Q: Can the slope of a line be used to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point and a sixth point and a seventh point?

A: Yes, the slope of a line can be used to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point and a sixth point and a seventh point. Choose seven points on the line and calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Then, use the equation y = mx + b and plug in the values of m and the coordinates of one of the points to find the equation of the line.

Q: How do I use the slope to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point and a sixth point and a seventh point and an eighth point?

A: To use the slope to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point and a sixth point and a seventh point and an eighth point, choose eight points on the line and calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Then, use the equation y = mx + b and plug in the values of m and the coordinates of one of the points to find the equation of the line.

Q: Can the slope of a line be used to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point and a sixth point and a seventh point and an eighth point and a ninth point?

A: Yes, the slope of a line can be used to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point and a sixth point and a seventh point and an eighth point and a ninth point. Choose nine points on the line and calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Then, use the equation y = mx + b and plug in the values of m and the coordinates of one of the points to find the equation of the line.

**Q: How do I use the slope to determine the equation of a line given a point and a slope and a second point and a third point and a fourth point and a fifth point