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Chapter 3: Problem 69

Write an equation for each line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. $$ (-2,-1) \text { and }(3,-4) $$

Short Answer

Expert verified
Slope-intercept form: \(y = -\frac{3}{5}x - \frac{11}{5}\). Standard form: \(3x + 5y = -11\).

Step by step solution

01

- Find the Slope

To find the equation of the line passing through two points, start by finding the slope. The formula for the slope (\text{m}) between two points \text{(x}_1\text{, y}_1\text{)} and \text{(x}_2\text{, y}_2\text{)} is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].Applying the given points (-2, -1) and (3, -4):\[ m = \frac{-4 - (-1)}{3 - (-2)} = \frac{-4 + 1}{3 + 2} = \frac{-3}{5} = -\frac{3}{5} \]
02

- Use the Point-Slope Form

Next, use the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] Choose one of the points, say (-2, -1), and substitute the slope (m) found in Step 1:\[ y - (-1) = -\frac{3}{5}(x - (-2)) \]Simplify:\[ y + 1 = -\frac{3}{5}(x + 2) \]
03

- Convert to Slope-Intercept Form

Now, convert the equation to slope-intercept form \(y = mx + b\)Solve for y:\[ y + 1 = -\frac{3}{5}(x + 2) \]Distribute the slope:\[ y + 1 = -\frac{3}{5}x - \frac{6}{5} \]Subtract 1 from both sides:\[ y = -\frac{3}{5}x - \frac{6}{5} - 1 \]Combine like terms:\[ y = -\frac{3}{5}x - \frac{11}{5} \]
04

- Convert to Standard Form

To convert the slope-intercept form to standard form \(Ax + By = C\), rearrange the equation:Start with:\[ y = -\frac{3}{5}x - \frac{11}{5} \]Multiply through by 5 to clear the fractions:\[ 5y = -3x - 11 \]Add 3x to both sides to get the standard form:\[ 3x + 5y = -11 \]

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Key Concepts

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

Finding Slope
To start understanding lines, we first find the slope. The slope tells us how steep the line is. Think of it as how many steps you go up or down (vertical change) for every step you go left or right (horizontal change). The formula for the slope (\text{m}) between two points \text{(x\(_1\), y\(_1\))} and \text{(x\(_2\), y\(_2\))} is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's use the given points (-2, -1) and (3, -4):
\[ m = \frac{-4 - (-1)}{3 - (-2)} = \frac{-4 + 1}{3 + 2} = \frac{-3}{5} = -\frac{3}{5} \]
This means for every 5 units we move to the right, the line goes down by 3 units.
Point-Slope Form
Next, we can use the point-slope form of a line's equation, which is very useful when we know a point on the line and the slope. The point-slope form is:
\[ y - y_1 = m(x - x_1) \]
Here, \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. Let's use the point (-2, -1) and the slope \(-\frac{3}{5}\) we found earlier:
\[ y - (-1) = -\frac{3}{5}(x - (-2)) \]
Simplify it:
\[ y + 1 = -\frac{3}{5}(x + 2) \]
This form shows how the coordinates change with respect to each other along the line.
Standard Form Equation
Finally, we convert our equation to the standard form. The standard form of a line鈥檚 equation is:
\[ Ax + By = C \]
Let's start from the slope-intercept form we derived:
\[ y = -\frac{3}{5}x - \frac{11}{5} \]
First, clear the fractions by multiplying everything by 5:
\[ 5y = -3x - 11 \]
Then, rearrange to bring all terms to one side and make it look like the standard form:
\[ 3x + 5y = -11 \]
Now we have the equation in standard form. Notice how the coefficients (\text{A, B}) and constant (\text{C}) are integers, making it easy to work with and interpret.

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Most popular questions from this chapter

Write an equation for each line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. $$ (-4,0) \text { and }(0,2) $$

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Solve each problem. The table gives heavy-metal nuclear waste (in thousands of metric tons) from spent reactor fuel stored temporarily at reactor sites, awaiting permanent storage. Let \(x=0\) represent \(1995, x=5\) represent 2000 (since \(2000-1995=5\) ), and so on. (Source: "Burial of Radioactive Nuclear Waste Under the Seabed, Scientific American.) \(\begin{array}{|c|c|}\hline \text {Yearx} & {\text { Waste } y} \\ {1995} & {32} \\ {2000} & {42} \\ {2010^{\star}} & {61} \\ {2020^{\star}} & {76} \\\ \hline\end{array}\) (a) For \(1995,\) the ordered pair is \((0,32) .\) Write ordered pairs for the data for the other years given in the table. (b) Plot the ordered pairs \((x, y) .\) Do the points lie approximately in a straight line? (c) Use the ordered pairs \((0,32)\) and \((25,76)\) to write the equation of a line that approximates the other ordered pairs. Give the equation in slope-intercept form. (d) Use the equation from part (c) to estimate the amount of nuclear waste in \(2015 .\) (Hint: What is the value of \(x\) for \(2015 ?\) )
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