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Chapter 4: Problem 33

Explain how to find an equation of a line when you are given two points on the line.

Short Answer

Expert verified
To find the equation of a line given two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\), first calculate the slope (m) using the formula \(m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\). Then, use the point-slope form of the line equation, \(y - y_{1} = m(x - x_{1})\), plugging the slope (m) and the coordinates of one of the given points. Finally, if necessary, rewrite the equation in slope-intercept form (y = mx + b) by solving for y: \(y = mx - m{x_1} + y_1\).

Step by step solution

01

Find the slope (m) of the line

To find the slope (m) of the line passing through the two given points, we use the slope formula: \[m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\] Plug in the coordinates of the two given points, and calculate the slope.
02

Use the point-slope form of the line equation

Now that we have the slope of the line (m), we can use the point-slope form of the line equation using one of the given points. The point-slope form of the line equation is: \[y - y_{1} = m(x - x_{1})\] Plug in the slope (m) from Step 1 and the coordinates of one of the given points, say \((x_{1}, y_{1})\).
03

Rewrite the equation in slope-intercept form, if necessary

You may need to rewrite the equation from Step 2 in slope-intercept form (y = mx + b) for easier interpretation. To do this, solve the equation from Step 2 for y: \[y = mx - m{x_1} + y_1\] Now you have the equation of the line in slope-intercept form, with m being the slope and b being the y-intercept (b = -m{x_1} + y_1). So, by following these steps, you can find the equation of a line given any two points on the line.

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Most popular questions from this chapter

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. \(18 x-3 y=9 ;(2,-2) ;\) slope-intercept form

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(-2,-3),(1,-3)\\\&L_{2}:(3,3),(3,5)\end{aligned}$$

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. perpendicular to \(2 x-6 y=-3\) containing \((2,2)\)

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=4\) and contains \((-4,-1)\)

Graph each function by making a table of values and plotting points. $$f(x)=\frac{2}{3} x+2$$
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