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Chapter 1: Problem 31

Find an equation of the line that passes through the given points. $$ (2,4) \text { and }(3,7) $$

Short Answer

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

Step by step solution

01

Calculate the slope of the line

To find the slope of the line, we will use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] Given points are \((2, 4)\) and \((3, 7)\), so we plug the values in the formula: \[m = \frac{7 - 4}{3 - 2} = \frac{3}{1}\] So, the slope of the line is \(m = 3\).
02

Use the point-slope form of the line

Now that we have the slope, we can use the point-slope form of the line to find the equation: \(y - y_1 = m(x - x_1)\). Choose one of the given points, for example \((2, 4)\), as \((x_1, y_1)\), and substitute the values in the point-slope form: \(y - 4 = 3(x - 2)\)
03

Simplify the equation

Now, we just need to simplify the equation and put it in slope-intercept form (\(y = mx + b\)), or any other form depending on the requirements of your exercise. Simplify the equation: \(y - 4 = 3x - 6\) Add \(4\) to both sides: \(y = 3x - 2\) And there we have it. The equation of the line passing through the points \((2, 4)\)and \((3, 7)\) is \(y = 3x - 2\).

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