Use a system of equations to solve each problem. See Example 8. Find an equation of the line y = ax + b that passes through the points (-2, 1) and (-1, -2).

Verified step by step guidance

1. Identify the coordinates of the two points through which the line passes. Let's denote these points as \\((x_1, y_1) = (-2, 1)\\) and \\((x_2, y_2) = (-1, -2)\\).
2. Use the slope formula \\((m = \frac{y_2 - y_1}{x_2 - x_1}\\)) to find the slope of the line. Substitute the coordinates of the points into the formula to calculate the slope.
3. With the slope calculated, use the point-slope form of the equation of a line, which is \\((y - y_1 = m(x - x_1)\\)), where \\((m\\)) is the slope and \\((x_1, y_1)\\) is one of the given points.
4. Substitute the slope and the coordinates of one of the points into the point-slope equation. Simplify the equation to solve for \\((y\\)) in terms of \\((x\\)).
5. Rearrange the equation into the slope-intercept form \\((y = ax + b\\)), where \\((a\\)) is the slope and \\((b\\)) is the y-intercept. This will give you the equation of the line in the desired form.

Key Concepts

System of Equations

A system of equations consists of two or more equations that share common variables. To solve such a system, one seeks values for the variables that satisfy all equations simultaneously. In this context, we will use the coordinates of the given points to create equations that represent the line's slope and y-intercept.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope and b the y-intercept. This form is particularly useful for graphing linear equations and understanding their behavior. In this problem, we need to determine the values of a (slope) and b (y-intercept) that define the line passing through the specified points.

Finding the Slope

The slope of a line measures its steepness and direction, calculated as the change in y divided by the change in x between two points. For the points (-2, 1) and (-1, -2), the slope can be found using the formula m = (y2 - y1) / (x2 - x1). This value will be crucial in forming the equation of the line in slope-intercept form.