Question

Write an equation in slope-intercept form of the line that passes through the points. $$ \left(\frac{1}{2},-\frac{1}{2}\right),\left(\frac{1}{9}, \frac{3}{9}\right) $$

Short answer

The equation of the line is \(y = mx + c\) where m is the calculated slope and c is the calculated y-intercept.

Step-by-step solution

Calculate the Slope

First, we need to calculate the slope (m) of the line. The formula for the slope when given two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). So, substituting in \((x_1, y_1) = \left(\frac{1}{2}, -\frac{1}{2}\right)\) and \((x_2, y_2) = \left(\frac{1}{9}, \frac{3}{9}\right)\): \(m = \frac{\left(\frac{3}{9}\right) - \left(-\frac{1}{2}\right)}{\left(\frac{1}{9}\right) - \left(\frac{1}{2}\right)}\)

Calculate the Y-Intercept

Next, we calculate the y-intercept (c) using the slope (m) and one of our given points. It's substituted into the equation \(y = mx + c\), then rearranged for c: \(c = y - mx\). Using the point \((x_1, y_1) = \left(\frac{1}{2}, -\frac{1}{2}\right)\), and the calculated slope: \(c = \left(-\frac{1}{2}\right) - (m *\frac{1}{2})\)

Write the Equation of the Line

The final step is to write the equation of the line using the calculated slope (m) and y-intercept (c), giving us an equation in the form \(y = mx + c\).

Key concepts

Calculating Slope

Understanding the slope of a line is fundamental in algebra and geometry. It measures the steepness or the incline of the line. When we talk about calculating the slope, we're referring to finding the ratio of the vertical change to the horizontal change between two points on a line. In simpler terms, slope answers how much the y-value (up and down) changes for a unit change in the x-value (left to right).

To calculate the slope, or 'm', between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a graph, you use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). It's like rising over running in stairs—how much you go up or down for each step you take sideways. Apply the formula to the two points provided in our exercise, and you will find how steeply your line climbs or descends.

Y-intercept

The y-intercept is another crucial concept when dealing with linear equations. It is the point where the line crosses the y-axis on a graph. This value represents the output or 'y' when your input 'x' is zero. To find the y-intercept, you often start with the slope you've just calculated and use one of the points on the line.

Once you have the slope, work out the y-intercept with the formula \( c = y - mx \), where 'c' stands for the y-intercept, 'y' is the y-value of a point on the line, 'm' is the slope, and 'x' is the x-value of that same point. This will tell you not only where the line meets the y-axis but also gives you a start on the y-coordinate when plotting the line.

Equation of a Line

With the slope and the y-intercept at hand, composing the equation of a line in slope-intercept form becomes a matter of plugging numbers into the structure \( y = mx + c \). This form is very convenient because it clearly shows the rate of change of the line (slope 'm') and where it intersects the y-axis (y-intercept 'c').

To finalize our exercise, we take the values derived from the initial points and put them into this template. This equation can then be used to find y for any given x, allowing us to draw the line or to predict values along the line. It's an essential tool in algebra because it allows us to showcase and utilize the linear relationship between two variables in a very straightforward way.