Problem 36 - Algebra and Trigonometry

Question

(a) Sketch the line with slope \(-2\) that passes through the point \((4,-1)\) . (b) Find an equation for this line.

Step-by-Step Solution

Answer

Equation: \( y = -2x + 7 \)

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of the equation of a line is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Our objective is to find this form of the equation for the line with a slope of \(-2\) and passing through the point \((4, -1)\).

Step 2: Plug in the Known Values

We have a slope \( m = -2 \) and a point \((x_1, y_1) = (4, -1)\) through which the line passes. Substitute these into the formula of a line to get \( -1 = -2(4) + b \).

Step 3: Solve for b, the Y-Intercept

Calculate \(-2(4)\), which gives \(-8\). Then substitute back: \( -1 = -8 + b \). Solve for \( b \) by adding \( 8 \) to both sides to find \( b = 7 \).

Step 4: Write the Equation of the Line

Now that we know \( b = 7 \), we can write the equation of the line as \( y = -2x + 7 \).

Step 5: Sketch the Line

To sketch the line, plot the y-intercept \((0, 7)\) and use the slope of \(-2\) to find another point. Starting at \((0, 7)\), move down 2 units and right 1 unit to reach \((1, 5)\). Draw a line through these points extending in both directions, and check that it passes through \((4, -1)\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a cornerstone concept in understanding linear equations. It's given by the formula \( y = mx + b \). Let's break this down:

• \(y\) represents the dependent variable, often referring to the vertical axis.
• \(x\) is the independent variable, usually associated with the horizontal axis.
• \(m\) is the slope, which describes the steepness and direction of the line.
• \(b\) represents the y-intercept, which is where the line crosses the y-axis.

In our exercise, the slope \(m\) is given as \(-2\). We need to fit this into the equation format by discovering the appropriate value for \(b\) so we can fully describe our line.

Finding Y-Intercept

Finding the y-intercept means identifying the point where the line crosses the y-axis, which is crucial in writing the full equation for a line. When a point is given, such as \(4,-1\), it helps us pinpoint that value:

• Starting with the slope-intercept form: \( y = mx + b \).
• Substitute the point (4, -1) into the equation: \( -1 = -2 \cdot 4 + b \).
• This simplifies to: \( -1 = -8 + b \).
• To solve for \(b\), add \(8\) to both sides: \( b = 7 \).

Thus, the y-intercept of our line is \(7\), completing our equation as \( y = -2x + 7 \). This step ensures our line accurately represents both its slope and intersection with the y-axis.

Graphing Lines

Graphing lines with precision involves using both the slope and y-intercept. Here's how it works:

• Begin by plotting the y-intercept on the graph. In our case, it's at \(0, 7\).
• The slope \(-2\) indicates the line goes down two units for every unit it goes right. From the y-intercept, move down 2 units and right 1 unit to locate a new point \(1, 5\).
• Each new point helps define the line. Continue plotting using the slope if needed to ensure accuracy.
• Once you have two or more points, draw a line through them, extending in both directions.

Verifying your line with the original point, such as confirming it passes through \(4, -1\), is a great way to ensure its correctness. Understanding graphing lines in this manner solidifies your grip on linear relationships in a visual format.