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Chapter 12: Problem 37

Find the equation of the line through the given points. $$(-2,-2) \text { and }(1,7)$$

Short Answer

Expert verified
The equation of the line that passes through the points \((-2,-2)\) and \((1,7)\) is \(y = 3x + 4\).

Step by step solution

01

Calculate the Slope

The first step is to calculate the slope of the line using the two points, \((-2,-2)\) and \((1,7)\). The formula to calculate the slope (m) is \(\frac{y2 - y1}{x2 - x1}\), where \(x1,y1\) and \(x2,y2\) are the coordinates of the two points. So, the slope is \(\frac{7 - (-2)}{1 - (-2)} = 3\).
02

Calculate the Y-intercept

The next step is to find the y-intercept. In order to calculate the y-intercept (c), use the formula \(c = y - mx\), where \(y\) and \(x\) are the coordinates of any of the given points, and \(m\) is the slope calculated in the previous step. Take \((-2, -2)\), therefore, \(c = -2 -3*(-2) = 4\).
03

Write the Equation of the Line

The final step is to write the equation of the line in slope-intercept form, \(y = mx + c\), using the slope (m) and y-intercept (c) that was calculated in the previous steps. Thus, the equation of the line is \(y = 3x + 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
Understanding the slope of a line is fundamental to grasping the concept of linear equations. Simply put, the slope measures the steepness or the tilt of a line. Imagine sliding down a hill; the slope tells you how quickly you would accelerate. Calculating the slope between two points like \( (-2, -2) \) and \( (1, 7) \) requires us to use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

This can be thought of as the 'rise over run', or how much the line goes up or down (\

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Most popular questions from this chapter

Explain how you would distinguish between the graphs of the two equations. a. \(y=\frac{3}{4} x+5\) b. \(y=\frac{3}{4} x-5\)

Suppose \(A\) and \(B\) are positive and \(C\) is negative. Is the point where the graph of \(A x+B y=C\) crosses the \(y\) -axis above or below the \(x\) -axis?

Suppose \(A\) and \(C\) are negative and \(B\) is positive. Is the point where the graph of \(A x+B y=C\) crosses the \(x\) -axis to the left or to the right of the \(y\) -axis?

Evaluate a function. Given \(f(x)=x^{2}-1,\) find \(f(1)\).

Evaluate a function. Given \(f(x)=\frac{x}{x+5},\) find \(f(-3)\).
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