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Chapter 11: Problem 236

Find the equation of the line passing through the points \((2,5)\) and \((6,2)\). Check the results graphically.

Short Answer

Expert verified
The equation of the line passing through the points \((2,5)\) and \((6,2)\) is \(y = -\frac{3}{4}x + \frac{13}{2}\).

Step by step solution

01

Find the slope of the line

To find the slope of the line, we will use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \) Here, \((x_1, y_1) = (2, 5)\) and \((x_2, y_2) = (6, 2)\). Now substitute these values into the formula: \( m = \frac{2 - 5}{6 - 2} \) \( m = \frac{-3}{4} \) So, the slope of the line is \( m = -\frac{3}{4} \).
02

Find the y-intercept

To find the y-intercept, we will substitute one of the given points (e.g., \((2,5)\)) into the equation of the line, and solve for the y-intercept (b). The equation is: \( y = mx + b \) Substituting the point \((2,5)\) and the slope \(m=-\frac{3}{4}\): \( 5 = -\frac{3}{4}(2) + b \) Now solve for b: \(5 = -\frac{3}{2} + b \) \(5 + \frac{3}{2} = b \) \(b = \frac{13}{2} \) Now we have the y-intercept \(b = \frac{13}{2}\).
03

Write the equation of the line

Now that we have the slope and the y-intercept, we can write the equation of the line: \(y = -\frac{3}{4}x + \frac{13}{2}\) This is the equation of the line that passes through the points \((2,5)\) and \((6,2)\).
04

Check the results graphically

To check the results graphically, we will plot the given points and our line on a coordinate plane: 1. Plot the points \((2, 5)\) and \((6, 2)\). 2. Draw the line with the equation \(y = -\frac{3}{4}x + \frac{13}{2}\). 3. Check to see if the line passes through both points. If the line passes through both points, then our solution is correct.

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Most popular questions from this chapter

Show that the graphs of \(3 x-y=9\) and \(6 x-2 y+9=0\) are parallel lines.

Find the slope, the \(\mathrm{y}\) -intercept, and the \(\mathrm{x}\) -intercept of the equation \(2 x-3 y-18=0\)

Write the equation in slope-intercept form; specify the slope and the \(y\) -intercept of the line. Sketch the graph of the equation. \(2 \mathrm{x}-3 \mathrm{y}=5\)

Graph the constant function \(2 \mathrm{y}=4\).

Find the equation of the line which passes through the points \((-3,1)\) and \((7,11)\)
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