Problem 47 - Algebra 1

Question

Write an equation of the line that passes through the points. (-3,1),(4,-2)

Step-by-Step Solution

Answer

So, the equation of the line passing through the given points is \(y = \frac{-3}{7}x + \frac{-2}{7}\)

Step 1: Calculate the Slope

To find the slope (m) of a line passing through two points \((-3,1)\) and \((4,-2)\), we use the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\], where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. So, \(m = \frac{-2 - 1}{4 - (-3)} = \frac{-3}{7}\).

Step 2: Use the Point-Slope Form

Once we've found the slope, we can use the point-slope form of a line to find the equation. The point-slope form is written as: \[y - y_1 = m(x - x_1)\]. Using one of the given points, for example \((-3,1)\), and the slope calculated in the previous step, we get: \[y - 1 = \frac{-3}{7}(x - (-3))\].

Step 3: Simplify Equation

Finally, simplify the equation to put it in the slope-intercept form (\(y = mx + b\)): \[y = \frac{-3}{7}x - \frac{9}{7} + 1 = \frac{-3}{7}x + \frac{-2}{7}\].

Key Concepts

Slope Calculation

Calculating the slope of a line involves determining its steepness or incline. This is done using the coordinates of two points that the line passes through. If you have two points, labeled as

• \((x_1, y_1)\) and
• \((x_2, y_2)\),

the formula to find the slope \(m\) is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula essentially measures the change in the vertical direction (known as rise) over the change in the horizontal direction (known as run).

For example, to find the slope of a line that passes through

• \((-3, 1)\) and
• \((4, -2)\),

substitute the coordinates into the slope formula:
\[ m = \frac{-2 - 1}{4 - (-3)} = \frac{-3}{7}\]
So, the slope of this line is \(-\frac{3}{7}\), indicating a gently descending line.

Point-Slope Form

Point-slope form is a useful way of expressing the equation of a line when you know both the slope and a single point on the line. The formula is:

• \(y - y_1 = m(x - x_1)\)

where

• \(m\) is the slope of the line, and
• \((x_1, y_1)\) is the specific point the line passes through.

Using point-slope form, you can easily derive an equation that represents your line. In our exercise, we've calculated the slope to be \(-\frac{3}{7}\), and we can use one of our given points, say \((-3,1)\).

Substitute these values into the formula:\[y - 1 = \frac{-3}{7}(x - (-3))\]
This undoubtedly yields an accurate starting equation for our line, which can be simplified further to express in other forms.

Slope-Intercept Form

The slope-intercept form is widely recognized for its practicality and straightforwardness. It expresses a linear equation as:

• \(y = mx + b\)

where:

• \(m\) represents the slope, and
• \(b\) is the y-intercept, the point where the line crosses the y-axis (when \(x = 0\)).

In our solution, after using the point-slope form, we simplified it to slope-intercept form. Beginning with:\[y - 1 = \frac{-3}{7}(x + 3)\]
Which simplifies to:\[y = \frac{-3}{7}x - \frac{9}{7} + 1\]Continuing:\[y = \frac{-3}{7}x + \frac{-2}{7}\]
Here, the equation \(y = \frac{-3}{7}x + \frac{-2}{7}\) efficiently describes the line's slope as \(-\frac{3}{7}\) and y-intercept as \(\frac{-2}{7}\), providing a complete and user-friendly representation of the linear relationship.