Find The Slope-intercept Form For The Line Satisfying The Conditions: Slope 6 7 \frac{6}{7} 7 6 ​ , Passing Through ( 5 , − 9 (5, -9 ( 5 , − 9 ].The Equation Is □ \square □ .

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Introduction

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The slope-intercept form of a line is a fundamental concept in mathematics, particularly in algebra and geometry. It is a way to express the equation of a line in the form $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept. In this article, we will find the slope-intercept form for the line satisfying the conditions: slope $\frac{6}{7}$, passing through $(5, -9)$.

Understanding the Slope-Intercept Form

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The slope-intercept form of a line is given by the equation $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept. The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run). In this case, the slope is given as $\frac{6}{7}$.

Finding the Equation of the Line

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To find the equation of the line, we can use the point-slope form of a line, which is given by the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. In this case, we are given the point $(5, -9)$ and the slope $\frac{6}{7}$.

Using the Point-Slope Form

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Using the point-slope form, we can substitute the values of the point and the slope into the equation:

$$y - (-9) = \frac{6}{7}(x - 5)$$

Simplifying the equation, we get:

$$y + 9 = \frac{6}{7}(x - 5)$$

Converting to Slope-Intercept Form

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To convert the equation to slope-intercept form, we need to isolate $y$ on one side of the equation. We can do this by subtracting $9$ from both sides of the equation:

$$y = \frac{6}{7}(x - 5) - 9$$

Expanding the right-hand side of the equation, we get:

$$y = \frac{6}{7}x - \frac{30}{7} - 9$$

Combining the constant terms on the right-hand side, we get:

$$y = \frac{6}{7}x - \frac{30 + 63}{7}$$

Simplifying the constant term, we get:

$$y = \frac{6}{7}x - \frac{93}{7}$$

Conclusion

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In this article, we found the slope-intercept form for the line satisfying the conditions: slope $\frac{6}{7}$, passing through $(5, -9)$. The equation of the line is $y = \frac{6}{7}x - \frac{93}{7}$. This equation represents a line with a slope of $\frac{6}{7}$ and a y-intercept of $-\frac{93}{7}$.

Final Answer

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The final answer is $\boxed{y = \frac{6}{7}x - \frac{93}{7}}$.

Discussion

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The slope-intercept form of a line is a fundamental concept in mathematics, and it is used to represent the equation of a line in a simple and intuitive way. In this article, we used the point-slope form of a line to find the equation of the line satisfying the conditions: slope $\frac{6}{7}$, passing through $(5, -9)$. The equation of the line is $y = \frac{6}{7}x - \frac{93}{7}$.

Related Topics

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• Slope-intercept form of a line
• Point-slope form of a line
• Equation of a line
• Slope of a line
• Y-intercept of a line

References

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• Slope-Intercept Form of a Line
• Point-Slope Form of a Line
• Equation of a Line
• Slope of a Line
• Y-Intercept of a Line
  

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Introduction

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In our previous article, we found the slope-intercept form for the line satisfying the conditions: slope $\frac{6}{7}$, passing through $(5, -9)$. In this article, we will answer some frequently asked questions related to the topic.

Q&A

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Q1: What is the slope-intercept form of a line?

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A1: The slope-intercept form of a line is a way to express the equation of a line in the form $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept.

Q2: How do I find the equation of a line using the point-slope form?

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A2: To find the equation of a line using the point-slope form, you need to substitute the values of the point and the slope into the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Q3: What is the difference between the slope-intercept form and the point-slope form?

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A3: The slope-intercept form is a way to express the equation of a line in the form $y = mx + b$, while the point-slope form is a way to express the equation of a line in the form $y - y_1 = m(x - x_1)$.

Q4: How do I convert the point-slope form to the slope-intercept form?

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A4: To convert the point-slope form to the slope-intercept form, you need to isolate $y$ on one side of the equation by subtracting $y_1$ from both sides and then simplifying the equation.

Q5: What is the y-intercept of a line?

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A5: The y-intercept of a line is the point where the line intersects the y-axis. It is represented by the value of $b$ in the slope-intercept form of a line.

Q6: How do I find the y-intercept of a line?

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A6: To find the y-intercept of a line, you need to substitute $x = 0$ into the equation of the line and solve for $y$.

Q7: What is the slope of a line?

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A7: 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).

Q8: How do I find the slope of a line?

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A8: To find the slope of a line, you need to use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Conclusion

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In this article, we answered some frequently asked questions related to the topic of finding the slope-intercept form for the line satisfying the conditions: slope $\frac{6}{7}$, passing through $(5, -9)$. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Answer

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The final answer is $\boxed{y = \frac{6}{7}x - \frac{93}{7}}$.

Discussion

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The slope-intercept form of a line is a fundamental concept in mathematics, and it is used to represent the equation of a line in a simple and intuitive way. In this article, we used the point-slope form of a line to find the equation of the line satisfying the conditions: slope $\frac{6}{7}$, passing through $(5, -9)$. The equation of the line is $y = \frac{6}{7}x - \frac{93}{7}$.

Related Topics

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• Slope-intercept form of a line
• Point-slope form of a line
• Equation of a line
• Slope of a line
• Y-intercept of a line

References

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• Slope-Intercept Form of a Line
• Point-Slope Form of a Line
• Equation of a Line
• Slope of a Line
• Y-Intercept of a Line