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Chapter 3: Problem 232

In the following exercises, find the equation of each line. Write the equation in slope-intercept form. Containing the points (-5,-3) and (4,-6)

Short Answer

Expert verified
\[y = -\frac{1}{3}x - \frac{14}{3}\]

Step by step solution

01

- Find the slope (m)

Use the slope formula to find the slope of the line passing through the points (-5, -3) and (4, -6). The slope formula is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the given points into the formula: \[m = \frac{-6 - (-3)}{4 - (-5)} = \frac{-6 + 3}{4 + 5} = \frac{-3}{9} = -\frac{1}{3}\]
02

- Use the point-slope form

The point-slope form of a line is: \[y - y_1 = m(x - x_1)\]Choose one of the given points, say (-5, -3), and substitute \(x_1 = -5\), \(y_1 = -3\), and \(m = -\frac{1}{3}\) into the point-slope form equation: \[y - (-3) = -\frac{1}{3}(x - (-5))\] \[y + 3 = -\frac{1}{3}(x + 5)\]
03

- Simplify to find the equation

Distribute \(-\frac{1}{3}\) and simplify the equation to get it into slope-intercept form. \[y + 3 = -\frac{1}{3}x - \frac{1}{3}(5)\] \[y + 3 = -\frac{1}{3}x - \frac{5}{3}\] Subtract 3 from both sides to isolate \(y\): \[y = -\frac{1}{3}x - \frac{5}{3} - 3\] Combine the constants: \[y = -\frac{1}{3}x - \frac{5}{3} - \frac{9}{3}\] \[y = -\frac{1}{3}x - \frac{14}{3}\]

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Key Concepts

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

slope-intercept form
The slope-intercept form of a linear equation is widely used because it lets us easily see both the slope and y-intercept of the line. The general formula is: \[ y = mx + b \] Here,
  • \( m \) represents the slope of the line, which is how steep the line is.
  • \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.
To find the slope-intercept form, you need the slope and the y-intercept. For a practical example, after finding the slope and using a point on the line, you substitute these into the point-slope formula and simplify to reach the slope-intercept form.
slope formula
The slope formula helps you determine the steepness or incline of a line between two points. The formula is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
To find the slope, you subtract the y-coordinate of the first point \(y_1\) from the y-coordinate of the second point \(y_2\), and divide it by the difference between the x-coordinates \(x_1\) and \(x_2\).
The slope tells you how much y changes for a unit change in x. For the points (-5, -3) and (4, -6), substituting the values gives: \[ m = \frac{-6 - (-3)}{4 - (-5)} = \frac{-3}{9} = -\frac{1}{3} \] The negative sign indicates the line slopes downwards as it moves from left to right.
point-slope form
Point-slope form is another way to write the equation of a line. It is particularly useful when you know one point on the line and the slope. The formula is: \[ y - y_1 = m(x - x_1) \] Here,
  • \( y_1 \) and \( x_1 \) are the coordinates of a known point on the line
  • \( m \) is the slope of the line
For example, using the point (-5, -3) and slope \( m = -\frac{1}{3} \), we get: \[ y + 3 = -\frac{1}{3}(x + 5) \] This form is a quick way to immediately write the equation of a line once you have a point and the slope.
line equation
To find the final form of the line equation in slope-intercept form from point-slope form, you'll need to simplify: Starting from the point-slope form:
\[ y + 3 = -\frac{1}{3}(x + 5) \]
Distribute \( -\frac{1}{3} \): \[ y + 3 = -\frac{1}{3}x - \frac{5}{3} \]
To isolate \( y \), subtract 3 from both sides: \[ y = -\frac{1}{3}x - \frac{5}{3} - 3 \] Combine constants: \[ y = -\frac{1}{3}x - \frac{14}{3} \] This is the slope-intercept form of the equation passing through the points (-5, -3) and (4, -6). The line equation helps us graph lines and understand how changes in one variable affect the other.

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Most popular questions from this chapter

In the following exercises, use the set of ordered pairs to ( ) determine whether the relation is a function, (b) find the domain of the relation, and \(\odot\) find the range of the relation. \([(-3,9),(-2,4),(-1,1),\) (0,0),(1,1),(2,4),(3,9)\\}

In the following exercises, (a) graph each function (b) state its domain and range. Write the domain and range in interval notation. $$ f(x)=3 x $$

Graph the linear inequality: \(x-y<4\)

Determine whether each ordered pair is a solution to the inequality \(3 x-4 y>4\) : (a) (5,1) (b) (-2,6) (c) (3,2) (d) (10,-5) (c) (0,0)

In the following exercises, (a) graph each function (b) state its domain and range. Write the domain and range in interval notation. $$ f(x)=-3 x^{2} $$
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