Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Containing the points (-3,4) and (2,5)

Short answer

The equation in slope-intercept form is \( y = \frac{1}{5}x + \frac{23}{5} \).

Step-by-step solution

- Find the slope

To find the slope of the line passing through the points (-3, 4) and (2, 5), use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the point coordinates into the formula: \[ m = \frac{5 - 4}{2 - (-3)} = \frac{1}{5} \]

- Use the point-slope form

Insert the slope \( m = \frac{1}{5} \) and one of the points, for example, (-3, 4), into the point-slope form equation: \[ y - y_1 = m(x - x_1) \] This results in: \[ y - 4 = \frac{1}{5}(x + 3) \]

- Simplify to slope-intercept form

Convert the equation from the point-slope form to the slope-intercept form \( y = mx + b \): \[ y - 4 = \frac{1}{5}x + \frac{3}{5} \] Then add 4 to both sides to obtain: \[ y = \frac{1}{5}x + \frac{3}{5} + 4 \] Combine the constants: \[ y = \frac{1}{5}x + \frac{3}{5} + \frac{20}{5} \] Thus, the equation in slope-intercept form is: \[ y = \frac{1}{5}x + \frac{23}{5} \]

Key concepts

slope formula

To find the slope of a line passing through two points, we use the slope formula. The slope is a measure of how steep the line is and is represented by the symbol \(m\). The formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] The numerator \(y_2 - y_1\) represents the change in the y-coordinates (vertical change), and the denominator \(x_2 - x_1\) represents the change in the x-coordinates (horizontal change). The division of these changes gives us the slope of the line. For example, to find the slope of the line through the points \((-3, 4)\) and \((2, 5)\), we substitute their coordinates into the formula: \[ m = \frac{5 - 4}{2 - (-3)} = \frac{1}{5} \] This tells us that the slope of the line is \(\frac{1}{5}\), meaning the line rises 1 unit for every 5 units it moves horizontally.

point-slope form

Once we know the slope of a line, we can use the point-slope form to write the equation of the line. The point-slope form of a line's equation is given by: \[ y - y_1 = m(x - x_1) \] Here, \(m\) is the slope of the line, and \((x_1, y_1)\) is any point on the line. By substituting the slope \(m = \frac{1}{5}\) and one of the points \((-3, 4)\) into the equation, we get: \[ y - 4 = \frac{1}{5}(x + 3) \] This equation now represents a line passing through the point \((-3, 4)\) with a slope of \(\frac{1}{5}\). It is a very useful form because it easily shows the relationship between the slope and a specific point on the line.

slope-intercept form

To make the line's equation simpler to use, we can convert it from point-slope form to slope-intercept form. This form is given by: \[ y = mx + b \] In this equation, \(m\) is the slope, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis. Starting with the point-slope form equation: \[ y - 4 = \frac{1}{5}(x + 3) \] We expand and simplify the equation: \[ y - 4 = \frac{1}{5}x + \frac{3}{5} \] Then, add 4 to both sides to isolate \(y\): \[ y = \frac{1}{5}x + \frac{3}{5} + 4 \] Combine the constant terms: \[ y = \frac{1}{5}x + \frac{3}{5} + \frac{20}{5} \] We get the final equation in slope-intercept form: \[ y = \frac{1}{5}x + \frac{23}{5} \] This form of the equation clearly shows the slope of the line \(\frac{1}{5}\) and the y-intercept \(\frac{23}{5}\), making it easy to graph and understand the linear relationship.