Problem 38 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ (5,-10),(12,-7) $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the given points is \(y = 3/7x -12.1428\)

Step 1: Calculation of the Slope

First, using the formula for the slope \((m)\) of a line passing through two points \((x1,y1)\) and \((x2, y2)\), we calculate that as follows: \(m = (y2-y1)/(x2-x1) = (-7 - (-10))/(12-5) = 3/7\). Thus, the slope of the line is \(3/7\).

Step 2: Calculation of the y-Intercept

Next, substitute the slope and one set of the coordinates into the slope-intercept form \((y=mx+c)\). Doing this gives us \(-10 = (3/7) * 5 + c\). Solving for \(c\) we get the value: \(c = -10 - 15/7 = -70/7 - 15/7 = -85/7 = -12.1428\).

Step 3: Write the Equation of the Line

Now, substitute the calculated slope and y-intercept into the slope-intercept form to get the equation of the line. After substitution, the equation of the line will be \(y = 3/7x -12.1428\).

Key Concepts

Equation of a Line

In mathematics, an equation of a line is a way to express the relationship between the coordinates of any point on that line. The most common form for this is the slope-intercept form, which is given by the equation \(y = mx + c\). Here, \(y\) and \(x\) are the coordinates of a point, \(m\) is the slope of the line, which tells us how steep the line is, and \(c\) is the y-intercept, where the line crosses the y-axis.

This form makes it easy to quickly understand and graph the line, as it provides both the rate of change (the slope) and the specific point where the line meets the y-axis. Understanding the equation of a line is fundamental in math as it helps illustrate how quantities are related in both a visual and algebraic way.

Slope Calculation

The slope, often represented by \(m\), is a measure of how much one quantity changes when another quantity changes. In the context of a line on a graph, it signifies how steep the line is. To compute the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

This formula calculates the ratio of the difference in the y-coordinates to the difference in the x-coordinates. For instance, for the points given in the example, \((5, -10)\) and \((12, -7)\), the slope is calculated as follows:

• \(m = \frac{-7 - (-10)}{12 - 5} = \frac{3}{7}\)

A positive slope means the line rises as it moves to the right, while a negative slope indicates the line falls as it moves to the right.

Y-Intercept Calculation

Once the slope \(m\) is determined, the next step is to find the y-intercept \(c\). The y-intercept is the value of \(y\) where the line intersects the y-axis. It gives you a starting point on the graph. To find \(c\), you can substitute the slope and the coordinates of one of the points into the slope-intercept equation \(y = mx + c\). For example, using the slope \(\frac{3}{7}\) and the point \((5, -10)\), you substitute into the equation as follows:

• \(-10 = \left(\frac{3}{7}\right) \cdot 5 + c\)
• Solve for \(c\) to get \(c = -10 - \frac{15}{7}\)
• \(c = \frac{-85}{7} \approx -12.1428\)

This calculation shows that the y-intercept of the line is approximately \(-12.1428\). By knowing both the slope and the y-intercept, you can fully write the equation of the line as \(y = \frac{3}{7}x - 12.1428\). This formula allows you to determine y-values for any given x, plotting the line accurately on a graph.