A Line Passes Through The Points \[$(-9, 8)\$\] And \[$(-6, 4)\$\] In The Standard \[$(x, Y)\$\] Coordinate Plane. The Equation Of The Line Is In The Form \[$y = Mx + B\$\].What Are The Values Of \[$m\$\] And

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Introduction

In the standard coordinate plane, a line can be represented by a linear equation in the form of {y = mx + b$}$, where {m$}$ is the slope and {b$}$ is the y-intercept. Given two points on the line, we can find the equation of the line using the slope-intercept form. In this article, we will explore how to find the equation of a line that passes through the points {(-9, 8)$}$ and {(-6, 4)$}$.

The Slope-Intercept Form

The slope-intercept form of a linear equation is given by {y = mx + b$}$, where {m$}$ is the slope and {b$}$ is the y-intercept. 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.

Finding the Slope

To find the slope of the line, we can use the formula {m = \frac{y_2 - y_1}{x_2 - x_1}$}$, where {x_1, y_1$}$ and {x_2, y_2$}$ are the coordinates of the two points on the line. In this case, the two points are {(-9, 8)$}$ and {(-6, 4)$}$.

# Define the coordinates of the two points
x1, y1 = -9, 8
x2, y2 = -6, 4

# Calculate the slope
m = (y2 - y1) / (x2 - x1)
print(m)

Calculating the Slope

Using the formula, we can calculate the slope as follows:

{m = \frac{4 - 8}{-6 - (-9)} = \frac{-4}{3} = -\frac{4}{3}$}$

Finding the Y-Intercept

Now that we have the slope, we can find the y-intercept by substituting one of the points into the equation {y = mx + b$}$. Let's use the point {(-9, 8)$}$.

# Define the slope and the coordinates of the point
m = -4/3
x, y = -9, 8

# Calculate the y-intercept
b = y - m * x
print(b)

Calculating the Y-Intercept

Using the point {(-9, 8)$}$, we can calculate the y-intercept as follows:

${8 = -\frac{4}{3}(-9) + b\$}

Simplifying the equation, we get:

${8 = 12 + b\$}

Subtracting 12 from both sides, we get:

{b = -4$}$

The Equation of the Line

Now that we have the slope and the y-intercept, we can write the equation of the line in the form {y = mx + b$}$.

# Define the slope and the y-intercept
m = -4/3
b = -4

# Print the equation of the line
print(f"y = {m}x + {b}")

The Final Equation

The final equation of the line is:

{y = -\frac{4}{3}x - 4$}$

Conclusion

Introduction

In our previous article, we explored how to find the equation of a line that passes through two points in the standard coordinate plane. We used the slope-intercept form of a linear equation and calculated the slope and y-intercept using the coordinates of the two points. In this article, we will answer some common questions related to finding the equation of a line.

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is given by {y = mx + b$}$, where {m$}$ is the slope and {b$}$ is the y-intercept.

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

A: To find the slope of a line, you can use the formula {m = \frac{y_2 - y_1}{x_2 - x_1}$}$, where {x_1, y_1$}$ and {x_2, y_2$}$ are the coordinates of the two points on the line.

Q: What is the y-intercept of a line?

A: The y-intercept of a line is the point where the line intersects the y-axis. It is represented by the value {b$}$ in the slope-intercept form of a linear equation.

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

A: To find the y-intercept of a line, you can substitute one of the points on the line into the equation {y = mx + b$}$ and solve for {b$}$.

Q: What is the equation of a line that passes through the points {(-3, 2)$}$ and {(1, 4)$}$?

A: To find the equation of the line, we can use the slope-intercept form of a linear equation. First, we need to find the slope using the formula {m = \frac{y_2 - y_1}{x_2 - x_1}$}$.

# Define the coordinates of the two points
x1, y1 = -3, 2
x2, y2 = 1, 4

# Calculate the slope
m = (y2 - y1) / (x2 - x1)
print(m)

Calculating the Slope

Using the formula, we can calculate the slope as follows:

{m = \frac{4 - 2}{1 - (-3)} = \frac{2}{4} = \frac{1}{2}$}$

Finding the Y-Intercept

Now that we have the slope, we can find the y-intercept by substituting one of the points into the equation {y = mx + b$}$. Let's use the point {(-3, 2)$}$.

# Define the slope and the coordinates of the point
m = 1/2
x, y = -3, 2

# Calculate the y-intercept
b = y - m * x
print(b)

Calculating the Y-Intercept

Using the point {(-3, 2)$}$, we can calculate the y-intercept as follows:

${2 = \frac{1}{2}(-3) + b\$}

Simplifying the equation, we get:

${2 = -\frac{3}{2} + b\$}

Adding {\frac{3}{2}$}$ to both sides, we get:

{b = 2 + \frac{3}{2} = \frac{7}{2}$}$

The Equation of the Line

Now that we have the slope and the y-intercept, we can write the equation of the line in the form {y = mx + b$}$.

# Define the slope and the y-intercept
m = 1/2
b = 7/2

# Print the equation of the line
print(f"y = {m}x + {b}")

The Final Equation

The final equation of the line is:

{y = \frac{1}{2}x + \frac{7}{2}$}$

Conclusion

In this article, we answered some common questions related to finding the equation of a line. We used the slope-intercept form of a linear equation and calculated the slope and y-intercept using the coordinates of the two points. The final equation of the line is {y = \frac{1}{2}x + \frac{7}{2}$}$.