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Chapter 1: Problem 19

Find the equation of the line through the given pair of points. Solve it for \(y\) if possible. $$(-1,-1),(3,4)$$

Short Answer

Expert verified
The equation of the line is \[ y = \frac{5}{4}x + \frac{1}{4} \].

Step by step solution

01

- Identify the Given Points

The given points are \((-1, -1)\) and \((3, 4)\). Let us denote these points as \((x_1, y_1)\) and \((x_2, y_2)\), respectively. Hence, we have \(x_1 = -1\), \(y_1 = -1\), \(x_2 = 3\), and \(y_2 = 4\).
02

- Find the Slope of the Line

The slope \(m\) of the line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the coordinates of the points, we get: \[ m = \frac{4 - (-1)}{3 - (-1)} = \frac{4 + 1}{3 + 1} = \frac{5}{4} \].
03

- Use the Point-Slope Form

The equation of a line with slope \(m\) passing through the point \((x_1, y_1)\) is given by the point-slope form: \(y - y_1 = m(x - x_1)\). Using \(m = \frac{5}{4}\) and the point \((-1, -1)\), we get: \[ y - (-1) = \frac{5}{4}(x - (-1)) \] \[ y + 1 = \frac{5}{4}(x + 1) \]
04

- Simplify to Solve for y

To solve the equation for \(y\), distribute the slope and isolate \(y\): \[ y + 1 = \frac{5}{4}x + \frac{5}{4} \] Subtract 1 from both sides: \[ y = \frac{5}{4}x + \frac{5}{4} - 1 \] Simplify the constant term: \[ y = \frac{5}{4}x + \frac{5}{4} - \frac{4}{4} = \frac{5}{4}x + \frac{1}{4} \]
05

Final Answer

The equation of the line in slope-intercept form is: \[ y = \frac{5}{4}x + \frac{1}{4} \]

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Key Concepts

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

Slope
The slope is a measure of how steep a line is. It's found by comparing the vertical change (rise) to the horizontal change (run) between two points on the line. For the points \((-1, -1)\) and \((3, 4)\), the formula for the slope, denoted by \(m\), is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in our points, we find: \[ m = \frac{4 - (-1)}{3 - (-1)} = \frac{4 + 1}{3 + 1} = \frac{5}{4} \] This tells us that for every 4 units we move horizontally, we move 5 units vertically.
Point-Slope Form
The point-slope form of a line's equation is very useful when we know a point on the line and the slope. This form is written as: \[ y - y_1 = m(x - x_1) \] Here, \(m\) is the slope and \( (x_1, y_1) \) is a specific point on the line. From our example: \[ y - (-1) = \frac{5}{4}(x - (-1)) \] Simplifying, we get: \[ y + 1 = \frac{5}{4}(x + 1) \] This form directly expresses the relationship between x and y using the known slope and point.
Slope-Intercept Form
The slope-intercept form is often the easiest to interpret and graph. It looks like this: \[ y = mx + b \] Here, \(m\) is the slope and \(b\) is the y-intercept, or where the line crosses the y-axis. We derive this form from the point-slope form by solving for \(y\). Starting from: \[ y + 1 = \frac{5}{4}(x + 1) \] We distribute \( \frac{5}{4} \) to get: \[ y + 1 = \frac{5}{4}x + \frac{5}{4} \] Subtract 1 from both sides to isolate \(y\): \[ y = \frac{5}{4}x + \frac{5}{4} - 1 \] Simplifying the constant term, we get: \[ y = \frac{5}{4}x + \frac{1}{4} \] Now we have a clear equation we can graph.
Linear Equations
A linear equation represents a straight line on a graph. Its general form is \(Ax + By + C = 0\), but it can be rearranged into forms that make certain features of the line more apparent. The slope-intercept form (\

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