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 of the line is \[ y = \frac{1}{5}x + \frac{23}{5} \].

Step-by-step solution

- Find the slope of the line

Use the formula for the slope \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \((x_1, y_1) = (-3, 4)\)and \((x_2, y_2) = (2, 5)\)\[ m = \frac{5 - 4}{2 + 3} = \frac{1}{5} \]So, the slope of the line is \(\frac{1}{5} \).

- Use the point-slope form to find the equation

The point-slope form of the equation of a line is\[ y - y_1 = m(x - x_1) \]Using the slope (\(m = \frac{1}{5}\)) and one of the points, say (-3, 4), the equation becomes\[ y - 4 = \frac{1}{5}(x + 3) \]

- Simplify to slope-intercept form

Now simplify the equation to get it into slope-intercept form (\( y = mx + b \)):\[ y - 4 = \frac{1}{5}(x + 3) \]\[ y - 4 = \frac{1}{5}x + \frac{3}{5} \]Add 4 to both sides:\[ y = \frac{1}{5}x + \frac{3}{5} + 4 \]Convert 4 to a fraction with a common denominator:\[ y = \frac{1}{5}x + \frac{3}{5} + \frac{20}{5} \]Combine the fractions:\[ y = \frac{1}{5}x + \frac{23}{5} \]

Key concepts

finding slope

The slope of a line indicates how steep the line is. To find the slope between two points, you can use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \( (x_1, y_1) = (-3, 4) \) and \( (x_2, y_2)= (2, 5) \). Plugging these values into the formula gives: \[ m = \frac{5 - 4}{2 + 3} = \frac{1}{5} \] So, the slope \(m\) is \(\frac{1}{5} \). The formula works by measuring the 'rise' (change in y) over the 'run' (change in x). Understanding the slope is crucial for graphing and interpreting lines.

point-slope form

The point-slope form of the equation of a line is very useful if you know one point on the line and the slope. The formula is: \[ y - y_1 = m(x - x_1) \] Using our slope \( m = \frac{1}{5} \) and one of the points, say \((-3, 4)\), we plug them into the formula: \[ y - 4 = \frac{1}{5}(x + 3) \] This format directly uses a point on the line and the slope to express the linear equation. It can be particularly handy when you need to quickly form an equation from a graph.

slope-intercept form

The slope-intercept form is perhaps the most commonly used form of a linear equation, easily recognized by its format: \[ y = mx + b \] 'm' represents the slope and 'b' represents the y-intercept (where the line crosses the y-axis). Starting from the point-slope form, \( y - 4 = \frac{1}{5}(x + 3) \), you simplify to slope-intercept form: \[ y - 4 = \frac{1}{5}x + \frac{3}{5} \] Adding 4 gives us: \[ y = \frac{1}{5}x + \frac{3}{5} + 4 \] Converting 4 to a common denominator: \[ y = \frac{1}{5}x + \frac{3}{5} + \frac{20}{5} \] Combining the fractions results in: \[ y = \frac{1}{5}x + \frac{23}{5} \] This form makes it simple to graph the line and see both the slope and y-intercept at a glance.