Problem 42 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. Through (5,-6) and (-6,5)

Step-by-Step Solution

Answer

The equation is \\(y = -x - 1\\).

Step 1: Determine the Slope

The slope of a line through two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let's substitute the coordinates (5,-6) and (-6,5) into this formula to calculate the slope. \ m = \frac{5 - (-6)}{-6 - 5} = \frac{11}{-11} = -1.\ So, the slope \(m\) is -1.

Step 2: Use the Point-Slope Formula

Once we have the slope, we can use the point-slope form of the equation of a line, which is \(y - y_1 = m(x - x_1)\). Let's take the point (5, -6) and substitute into the formula along with the slope \(m = -1\): \ y + 6 = -1(x - 5) \

Step 3: Simplify to Slope-Intercept Form

Finally, we need to convert the equation from point-slope form to slope-intercept form \(y = mx + b\). Distribute the slope on the right-hand side: \ y + 6 = -x + 5.\ Then, subtract 6 from both sides to isolate y: \ y = -x - 1.\ So, the equation of the line in slope-intercept form is \ y = -x - 1.\

Key Concepts

Slope Calculation

When it comes to finding the equation of a line through two points, calculating the slope is the first step on your journey. The slope, often represented by the letter \(m\), tells us how steep a line is and the direction it goes (uphill, downhill, or flat).

To find the slope, we use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points on the line.

In the example from the exercise, the points were \((5, -6)\) and \((-6, 5)\). Substituting these into the slope formula:

• The change in \(y\) (vertical difference) is \(5 - (-6) = 11\).
• The change in \(x\) (horizontal difference) is \(-6 - 5 = -11\).

The slope \(m\) becomes \(\frac{11}{-11} = -1\).

A negative slope, like \(-1\), indicates the line goes downward from left to right. Understanding slope is fundamental because it gives you a big clue about how the line behaves.

Point-Slope Form

Once you have the slope, the next step is to use the point-slope form to express the line's equation. The point-slope form of a line is expressed as \(y - y_1 = m(x - x_1)\). This equation is particularly useful because it plugs directly into a point on the line and the slope we just calculated.

It's a powerful tool that always holds true, whether your line is vertical, horizontal, or at a slant.

In the exercise's example, we used the point \((5, -6)\) and the slope \(-1\). By substituting these values into our formula, we get:

\(y + 6 = -1(x - 5)\).

This form is very flexible and can easily be re-arranged to other forms, such as the slope-intercept form.

The point-slope form is like a bridging concept, connecting your understanding of the relationship between points and the algebraic representation of a line. It's especially handy when you're given a point and the slope, offering a straightforward way to get to the equation of the line.

Equation of a Line

The equation of a line can be expressed in various forms, but the slope-intercept form \(y = mx + b\) is one of the most popular. This format gives you two key pieces of information about the line at a glance: the slope \(m\) and the y-intercept \(b\).

To derive the slope-intercept form from the point-slope form, you simply need to isolate \(y\) on one side of the equation.

For our line with point-slope form \(y + 6 = -1(x - 5)\), follow these steps:

• Distribute the slope on the right side: \(y + 6 = -x + 5\).
• Then subtract 6 from both sides to solve for \(y\): \(y = -x - 1\).

Here, the slope \(-1\) confirms a downward line from left to right, while \(-1\) is the point where the line crosses the y-axis.

The slope-intercept form is practical for graphing because it immediately shows those two essential characteristics of the line. It's an elegant way to summarize what you've learned about a given line's direction and intercept.