Problem 27 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. \(x\) intercept \(1 ; \quad y\) intercept \(-3\)

Step-by-Step Solution

Answer

The equation of the line is \( y = 3x - 3 \).

Step 1: Identify Intercepts

The problem gives you the x-intercept as 1 and the y-intercept as -3. Recall that at the x-intercept, the y-coordinate is 0 and at the y-intercept, the x-coordinate is 0. Therefore, the x-intercept point is (1, 0) and the y-intercept point is (0, -3).

Step 2: Determine the Slope

We use the formula for the slope, which is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the points (1, 0) and (0, -3) into this formula: \( m = \frac{0 - (-3)}{1 - 0} = \frac{3}{1} = 3 \). So, the slope \( m \) is 3.

Step 3: Use the Slope-Intercept Form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We have the slope \( m = 3 \) and the y-intercept \( b = -3 \). Substitute these values into the equation to get \( y = 3x - 3 \).

Step 4: Write the Equation

Having calculated all required components, the equation of the line with an x-intercept of 1 and a y-intercept of -3 is \( y = 3x - 3 \).

Key Concepts

Intercepts

Intercepts are key points on a graph where a line crosses the axes. Understanding intercepts helps in determining the path of a line through the coordinate plane.

An **x-intercept** is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. For example, if given the x-intercept as 1, the point can be written as (1,0). This means at this location on the graph, the line meets the x-axis.

A **y-intercept** is where the line meets the y-axis. Here, the x-coordinate equals zero. For example, with a y-intercept at -3, this point becomes (0,-3). The line will cross the y-axis at this point.

• Knowing both intercepts offers a clear picture of where the line initially crosses both axes.
• Intercepts are crucial for forming linear equations as they provide fixed points to work with.

By plotting these intercepts, you can start forming the path of the line on a graph.

Slope

The slope is a measure of the steepness or tilt of a line. It tells us how much the line rises or falls as we move from left to right across the graph.

Using two points, (1,0) and (0,-3) as established from the intercepts, calculate the slope (`m`) with the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In this example, the calculation is \( m = \frac{0 - (-3)}{1 - 0} = \frac{3}{1} = 3 \). With a slope of 3:

• The line rises 3 units vertically for each unit it moves horizontally.
• This positive slope suggests an upward movement from left to right.

The slope is crucial for understanding the extent of a line's inclination or declination on the graph.

Equation of a Line

The equation of a line provides a complete mathematical representation of the line's behavior on a graph. Using the slope-intercept form is one of the most straightforward ways to write this equation.

In the **slope-intercept form**, \( y = mx + b \), \( m \) represents the slope, and \( b \) denotes the y-intercept. With the slope calculated as 3 and the y-intercept given as -3, the equation integrates these values as:

\[ y = 3x - 3 \]

This equation effectively describes our line using two main components:

• **Slope (\( m = 3 \))** ensures the line moves upward at a steady pace.
• **Y-intercept (\( b = -3 \))** decides where the line will intersect the y-axis.

Armed with the full equation, you can graph this line and predict its behavior extensively across the coordinate grid.