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Chapter 1: Problem 33

Find an equation of the line that passes through the given points. $$ (1,2) \text { and }(-3,-2) $$

Short Answer

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

Step by step solution

01

Find the slope of the line.

To find the slope of the line that passes through the given points \((1, 2)\) and \((-3, -2)\), we will use the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) where \(m\) is the slope, and \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two given points. In our case, we have \(x_1 = 1\), \(y_1 = 2\), \(x_2 = -3\), and \(y_2 = -2\). Plugging in these values, we get: \(m = \frac{-2 - 2}{-3 - 1} = \frac{-4}{-4} = 1\)
02

Use the point-slope form.

Now that we have the slope \(m = 1\), we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by: \(y - y_1 = m(x - x_1)\) We can use either of the given points for our calculation, so let's use the point \((1, 2)\). Plugging the values into the equation, we get: \(y - 2 = 1(x - 1)\)
03

Simplify the equation.

Now let's simplify the equation we obtained in step 2 by distributing the slope on the right side and solving for \(y\): \(y - 2 = 1(x - 1)\) \(y - 2 = x - 1\) \[y = x + 1\] So, the equation of the line passing through the given points is \(y = x + 1\).

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Most popular questions from this chapter

Write the equation in the slopeintercept form and then find the slope and \(y\) -intercept of the corresponding line. $$ x-2 y=0 $$

Sketch the straight line defined by the linear equation by finding the \(x\) - and \(y\) -intercepts. $$ 2 x-5 y+10=0 $$

If the slope of the line \(L_{1}\) is positive, then the slope of a line \(L_{2}\) perpendicular to \(L_{1}\) may be positive or negative.

Metro Department Store's annual sales (in millions of dollars) during the past 5 yr were. $$ \begin{array}{lccccc} \hline \text { Annual Sales, } y & 5.8 & 6.2 & 7.2 & 8.4 & 9.0 \\ \hline \text { Year, } \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 \\ \hline \end{array} $$ a. Plot the annual sales \((y)\) versus the year \((x)\). b. Draw a straight line \(L\) through the points corresponding to the first and fifth years. c. Derive an equation of the line \(L\). d. Using the equation found in part (c), estimate Metro's annual sales 4 yr from now \((x=9)\).

Is there a difference between the statements "The slope of a straight line is zero" and "The slope of a straight line does not exist (is not defined)"? Explain your answer.
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