Problem 21 - Algebra 2

Question

Graph the line passing through the given point with the given slope. $$ (3,-4), m=2 $$

Step-by-Step Solution

Answer

Graph line using slope-intercept form: \(y = 2x - 10\).

Step 1: Understand the Problem

We need to graph a line that passes through the point \((3, -4)\) with a slope \(m = 2\).

Step 2: Recall the Point-Slope Form Equation

The point-slope form of a line's equation is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope.

Step 3: Plug in Known Values

Substitute the known point \((3, -4)\) and slope \(m = 2\) into the equation: \[y - (-4) = 2(x - 3)\] This simplifies to \[y + 4 = 2(x - 3)\].

Step 4: Simplify the Equation to Slope-Intercept Form

First, distribute the 2 on the right side of the equation: \[y + 4 = 2x - 6\]. Now, isolate \(y\) by subtracting 4 from both sides: \[y = 2x - 10\]. This is the slope-intercept form \(y = mx + b\), where \(b\) is the y-intercept.

Step 5: Graph the Equation

Start by plotting the point \((3, -4)\) on a graph. Use the slope \(m = 2\), which means rise 2 and run 1 from the point \((3, -4)\), to find another point. Continue plotting points and draw a straight line through them. Alternatively, start at the y-intercept \(b = -10\) and follow the slope to form the line.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation helps us emphasize the slope of the line directly from a specific point. This form is useful when you know one point on the line and want to easily unravel other related aspects. The general formula is expressed as \( y - y_1 = m(x - x_1) \), where:

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

This format directly connects the difference in y-values to the difference in x-values, making it a dynamic way to write the equation when you start with a point and a slope. To apply this in our example, for the point \((3, -4)\) with a slope \(m = 2\), insert these values into the formula. This step offers a concrete way to begin graphing a linear function since it naturally ties these two vital elements together, setting a foundation for further simplification if needed.

Slope-Intercept Form

The slope-intercept form is perhaps the most familiar form of a linear equation. It provides an efficient view of how the line behaves and intersects various axes. This form is given by \( y = mx + b \), where:

• \( m \) is the slope of the line, showing how much \( y \) increases with a unit increase in \( x \).
• \( b \) is the y-intercept, signifying where the line crosses the y-axis.

Through this formula, you see a direct link between the slope and the starting point \(y\)-wise on the graph. In our exercise, converting the equation from point-slope \( y + 4 = 2(x - 3) \) to slope-intercept gives \( y = 2x - 10 \). By doing so, we clearly identify that as \( x \) increases by one, \( y \) climbs by 2, and also that the line intersects the y-axis at \( -10 \). With the simplicity it offers, understanding the basic construction and transformation into this form makes interpretation and graphing straightforward.

Graphing Lines

Graphing lines essentially embodies taking theoretical equations into a visual realm. For completing this task, we use either of the forms discussed previously to help illustrate the line on a coordinate plane. Here’s how it's usually done:

• Start with a known point. For example, plot \((3, -4)\).
• Utilize the slope, \( m = 2 \), as a guide. From \((3, -4)\), rise 2 and run 1 to determine new points.
• Alternatively, begin at the y-intercept from the slope-intercept form, which is \(-10\), and apply the slope to trace the line.

By following these steps, you’ll plot multiple points, providing a clearer view of the line's direction and inclination, all while reinforcing the underlying rule it obeys: every point on the line satisfies that linear equation. This process helps demystify how equations translate into straight lines and contribute to a fuller understanding of algebraic graphing principles.