Problem 24 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Through \((-1,-2)\) and \((4,3)\)

Step-by-Step Solution

Answer

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

Step 1: Find the slope

To find the slope of the line, use the slope formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the given points \((x_1, y_1) = (-1, -2)\) and \((x_2, y_2) = (4, 3)\): \[m = \frac{3 - (-2)}{4 - (-1)} = \frac{5}{5} = 1\]Therefore, the slope \(m\) of the line is 1.

Step 2: Use point-slope form

The point-slope form of a line's equation is given by \[y - y_1 = m(x - x_1)\]Using the slope \(m = 1\) and the point \((x_1, y_1) = (-1, -2)\), the equation becomes: \[y - (-2) = 1(x + 1)\]Simplifying, we have:\[y + 2 = x + 1\]

Step 3: Convert to slope-intercept form

Subtract 2 from both sides to solve for \(y\):\[y = x + 1 - 2\]Simplify the equation:\[y = x - 1\]Now the equation is in slope-intercept form \(y = mx + b\), where \(m = 1\) and \(b = -1\).

Key Concepts

Slope Formula

The slope formula is an essential concept in understanding how steep a line is on a graph. The slope describes how much the dependent variable (usually represented by \(y\)) changes for a unit change in the independent variable (usually represented by \(x\)). In simpler terms, it's the rise over the run. The formula itself is given by:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.The slope \(m\) is a measure of the line's inclination. When you plug in the coordinates of two points, you calculate how one moves vertically (in the \(y\) direction) compared to horizontally (in the \(x\) direction). For example, if calculating the slope using points \((-1, -2)\) and \((4, 3)\), as shown in the exercise, you'd perform the following calculation:

• \(m = \frac{3 - (-2)}{4 - (-1)} = \frac{5}{5} = 1\)

Thus, the slope \(m\) is 1, indicating a positive slope where for every unit increase in \(x\), \(y\) also increases by one unit. This consistency continues across the graph.

Point-Slope Form

Once you know the slope, the point-slope form helps you establish an equation for the line using any point on that line. This form is especially useful when you have a point and a slope readily available. In the form:

• \(y - y_1 = m(x - x_1)\)

\((x_1, y_1)\) is a specific point through which the line passes. The "\(m\)" represents the slope we just found. For instance, utilizing the point \((-1, -2)\) and the slope \(m = 1\), the point-slope equation becomes:

• \(y - (-2) = 1(x - (-1))\)

Simplifying, it converts to:

• \(y + 2 = x + 1\)

This stage creates a foundation to adjust into another popular line equation format, showing the step-by-step algebraic journey.

Slope-Intercept Form

The slope-intercept form is probably the most recognized line equation format, as it is straightforward to interpret. This form breaks down into:

• \(y = mx + b\)

"\(m\)" is the slope, and \(b\) is the y-intercept - the point where the line pierces the \(y\)-axis.In our exercise, after using the point-slope form and simplifying, the aim is to express \(y\) straightforwardly. Starting with:

• \(y + 2 = x + 1\)

Subtracting 2 from both sides provides:

• \(y = x - 1\)

This result shows the slope \(m = 1\) and the y-intercept \(b = -1\). The equation \(y = x - 1\) is in slope-intercept form, clearly highlighting the line mechanics and enabling quick reading of the line's characteristics straight from the equation.