Problem 32 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-3,6)\) and \((3,-2)\)

Step-by-Step Solution

Answer

The equation for the line in point-slope form is \(y-6=-1(x+3)\) or \(y-6=-x-3\), and in slope-intercept form it is \(y = -x + 3\).

Step 1: Find the Slope

The first step is to find the slope of the line. The formula to calculate the slope or gradient between two points, say \((-3,6)\) and \((3,-2)\), is given by \[m = \frac{(y2-y1)}{(x2-x1)}\]. So, computing this gives us the value of the slope: \[m = \frac{(-2-6)}{(3-(-3))} = -1\].

Step 2: Write the equation in point-slope form

Once the slope is known, we substitute it and one of the points into the point-slope formula. Let's use the point \((-3,6)\): \(y-y1=m(x-x1)\) becomes \(y-6=-1(x-(-3))\), which simplifies to \(y-6=-1(x+3)\). It's also possible to express it in expanded form, giving \(y-6=-x-3\).

Step 3: Write the equation in slope-intercept form

To find the equation in slope-intercept form, we will solve the point-slope form equation for \(y\): \(y = -1(x+3) + 6\). This simplifies to \(y = -x - 3 + 6\), which in turn simplifies to \(y = -x + 3\).

Key Concepts

Linear Equations

Linear equations represent straight lines on a graph. A linear equation in two variables, such as \(x\) and \(y\), depicts a linear relationship between them.
These equations take the form \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.

• The slope \(m\) describes how steep the line is.
• The y-intercept \(b\) is the point where the line crosses the y-axis.

Given two points, such as in this exercise,

• the first step is finding the slope using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
• This calculation helps us write the equation in different forms.

Mastering linear equations allows us to model many real-world situations, from predicting trends to calculating costs.

Slope in Point-Slope Form

When we have a slope and a point, the point-slope form is quite handy. This form is written as \(y - y_1 = m(x - x_1)\), where:

• \((x_1, y_1)\) is a known point on the line.
• \(m\) is the slope of the line.

In this exercise, after we found the slope to be \(-1\), substituting the point \((-3,6)\) gives us:

• \(y - 6 = -1(x + 3)\).

This formula allows us to easily find a line equation if the slope and at least one point is known. By substituting this formula, we can transform any line equation into other forms, such as the slope-intercept form, while maintaining the same slope and points.

Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\). It explicitly shows the slope \(m\) and the y-intercept \(b\). Converting our point-slope form, \(y - 6 = -1(x + 3)\), into this form involves solving for \(y\):

• Expand the equation: \(y - 6 = -x - 3\).
• Solve for \(y\): \(y = -x - 3 + 6\).
• Simplify further: \(y = -x + 3\).

Thus, the line equation becomes \(y = -x + 3\),

• where \(-x\) shows the slope.
• \(+3\) indicates the y-intercept, the point where the line crosses the y-axis.

This form is often preferred when you need to quickly see both the slope and the y-intercept of the line, providing a clear and immediate understanding of the line's behavior.