Problem 47 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of line that passes through the given points. \((0,-3)\) and \((6,5)\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = \frac{4}{3}x -3\).

Step 1: Find the slope

The slope can be found using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points it becomes: \(m = \frac{5 - (-3)}{6 - 0} = \frac{8}{6} = \frac{4}{3}\). So, the slope m is \(\frac{4}{3}\).

Step 2: Find the y-intercept

The y-intercept, b, can be solved with the slope point form of a line (y - y1 = m(x - x1)). Substitute the slope found and one of the points into the equation to solve for b. For point (0,-3) it is: \(-3 - 0 = \frac{4}{3}(0 - 0) => -3 = b\). So, the y-intercept is -3.

Step 3: Formulate the Equation of Line

We substitute the values of m and b we've inferred into the slope-intercept form y = mx + b, yielding the equation of the line: \(y = \frac{4}{3}x -3\)

Key Concepts

Finding Slope

The slope of a line measures its steepness and direction and is an essential ingredient when it comes to understanding linear equations. To calculate the slope between two points, like (0,-3) and (6,5), we use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula subtracts the 'y' values and divides by the difference in 'x' values of the two points. For the points provided, the slope (m) is calculated as \(m = \frac{5 - (-3)}{6 - 0} = \frac{8}{6} = \frac{4}{3}\), which simplifies to \(\frac{4}{3}\).

A positive slope, like the one we found here, indicates that the line rises as it moves from left to right. A negative slope would mean the line falls as it moves in the same direction. If the slope is zero, the line is horizontal, and if the slope is undefined, it means the line is vertical.

Y-intercept

The y-intercept is the point where the line crosses the y-axis. It is represented by the letter 'b' in the slope-intercept form of a line, which is \(y = mx + b\). In our problem, finding the y-intercept is pretty straightforward since one of the given points is (0,-3), which already lies on the y-axis. Therefore, the y-intercept (b) is -3.

When a point has an 'x' value of 0, it will always be located on the y-axis, making its 'y' value the y-intercept of the line. This simplifies our work, as no additional steps are needed to find 'b' when such a point is given.

Equation of a Line

The equation of a line provides a formula that relates all points along that line. For lines in a two-dimensional space, the most frequently used form is the slope-intercept form, given by \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.

In our example, having already calculated the slope \(m = \frac{4}{3}\) and identifying the y-intercept as (b = -3), we can construct the linear equation. By placing these values into the slope-intercept formula, we get \(y = \frac{4}{3}x - 3\).

This equation now represents a set of instructions for creating the line: for any value of 'x', multiply it by \(\frac{4}{3}\) and then subtract 3 to find the corresponding 'y' value. This will place a point on the line described by this equation.

Slope-Point Form

Apart from the slope-intercept form, there's another useful equation format known as the slope-point form, or point-slope form, which is particularly handy when we have a slope and a specific point the line passes through. The formula for this form is \(y - y_1 = m(x - x_1)\), where (m) is the slope and \((x_1, y_1)\) is the point on the line.

To use this equation, you simply fill in the values of the slope and the coordinates of the given point. For example, if we chose point (0,-3) from our pair of points, we already have the slope \(m = \frac{4}{3}\), and putting these into the formula we get \(-3 - 0 = \frac{4}{3}(0 - 0)\). Although in this case, the calculation simplifies because of the zeros, this form is extremely useful for working with any point-slope combinations.