Problem 59 - Algebra 1

Question

Write an equation of the line in slope-intercept form that passes through the two points, or passes through the point and has the given slope. $$(6,5),(2,1)$$

Step-by-Step Solution

Answer

The equation of the line passing through the points (6,5) and (2,1) in slope-intercept form is \(y = -x + 3\)

Step 1: Determine the slope

The slope, m, of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\) . Plugging in the two given points gives \(m = \frac{1 - 5}{2 - 6} = -1 \)

Step 2: Use point-slope form

Use the calculated slope and one of the given points, say (2,1), to write the equation in point-slope form, \(y - y_1 = m(x - x_1)\), so the equation becomes \(y - 1 = -1(x - 2)\)

Step 3: Convert to slope-intercept form

To get the equation in slope-intercept form (y = mx + c), distribute the -1 on the right-hand side and add 1 to both sides, which gives \(y = -x + 3\)

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is. It is often represented by the letter \( m \). Calculating the slope involves using two points on the line. These points can be labeled as \((x_1, y_1)\) and \((x_2, y_2)\). The formula for finding the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula finds the rate of change of the vertical distance (change in \( y \)) with respect to the horizontal distance (change in \( x \)).
For example, if you are given the points \((6, 5)\) and \((2, 1)\), plug these into the formula:

• Change in \( y \): \(y_2 - y_1 = 1 - 5 = -4\)
• Change in \( x \): \(x_2 - x_1 = 2 - 6 = -4\)

Thus, the slope \( m \) is \(-1\). A negative slope indicates that as you move from left to right, the line falls.

Point-Slope Form

The point-slope form is a way to write the equation of a line. It makes use of the slope and a specific point on the line. This form is written as:\[ y - y_1 = m(x - x_1) \]Where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. This form is particularly useful when you know the slope and at least one point on the line.
For instance, given the point \((2, 1)\) and a slope of \(-1\), substitute these into the formula to get:

• \(y - 1 = -1(x - 2)\)

This equation represents the line in point-slope form. It allows us to efficiently plug in any point and the slope to find the equation of the line.

Linear Equations

Linear equations are equations that graph as straight lines on the coordinate plane. They can come in different forms, with two of the most common being the point-slope form and the slope-intercept form.The equation of a line can be converted to slope-intercept form, which is written \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.
Using the point-slope form \( y - 1 = -1(x - 2) \) from the previous example, you can change it to slope-intercept form by simplifying:

• Distribute: \( y - 1 = -x + 2 \)
• Add 1 to both sides: \( y = -x + 3 \)

In this equation, we see that the slope \( m \) is \(-1\) and the y-intercept \( b \) is \(3\). This form is useful for quickly identifying the slope and the starting point of the line at the y-axis. Understanding these forms helps simplify graphing and working with linear equations across a variety of mathematical problems.