Problem 27 - Precalculus Mathematics for Calculus

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: Understand the Intercepts

The given x-intercept is 1. This means the line passes through the point \((1, 0)\). The y-intercept is -3, so the line passes through the point \((0, -3)\).

Step 2: Calculate the Slope

The slope \(m\) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute \((x_1, y_1) = (1, 0)\) and \((x_2, y_2) = (0, -3)\) into the formula to find the slope:\[m = \frac{-3 - 0}{0 - 1} = \frac{-3}{-1} = 3\].

Step 3: Use Point-Slope Form

Now that we have the slope \(m = 3\), we can use the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\). Using point \((1, 0)\), the equation becomes:\[y - 0 = 3(x - 1)\].

Step 4: Simplify the Equation

Simplify the equation from the point-slope form. Start with \[y = 3(x - 1)\]. Then distribute the 3: \[y = 3x - 3\]. This is the equation of the line in slope-intercept form \(y = mx + b\).

Step 5: Verify by Checking Intercepts

Lastly, check the x-intercept by setting \(y = 0\) and solving for \(x\) in the equation \(3x - 3 = 0\). Solving gives \(x = 1\). Check the y-intercept by setting \(x = 0\) which gives \(y = -3\). Both intercepts match, confirming the solution.

Key Concepts

Understanding the X-Intercept

The **x-intercept** of a line is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. In simpler terms, it's where the graph touches or crosses the horizon of the x-axis. This point is vital because it helps in sketching the graph of the equation or understanding properties of the line. To find the x-intercept in an equation, set the y-value to zero and solve for x. For example, if given an equation like \( y = 3x - 3 \), set \( y = 0 \) to find:

• \( 0 = 3x - 3 \)
• Solving the equation, \( 3x = 3 \)
• \( x = 1 \)

Hence, the x-intercept is 1, confirmed as the point \((1, 0)\). Knowing how to find this point is crucial when determining the equation of a line from given conditions.

Understanding the Y-Intercept

The **y-intercept** of a line is the point where the line crosses the y-axis. At this location, the x-coordinate is always zero. This point tells us the position of the line relative to the vertical axis and is a key starting point for graphing.To determine the y-intercept, you'd set the value of x to zero in the line's equation and solve for y. For instance, consider the equation \( y = 3x - 3 \). When we set \( x = 0 \), it becomes:

• \( y = 3(0) - 3 \)
• \( y = -3 \)

Thus, the y-intercept is -3, represented as the point \((0, -3)\). These intercepts are not just numbers; they offer a way to visualize the line in a coordinate plane.

Slope-Intercept Form

The **slope-intercept form** of a line's equation is expressed as \( y = mx + b \). This format is exceptionally useful for quickly identifying the slope and y-intercept of the line:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept.

In our example, the line’s equation became \( y = 3x - 3 \). Here:

• The slope \( m \) is 3.
• The y-intercept \( b \) is -3.

This form simplifies not just the mathematics, but it also makes drawing the line straightforward on a graph. Start at the y-intercept, and use the slope to determine how the line ascends or descends across the graph.

Point-Slope Form

The **point-slope form** of a line is another useful way to express the equation of a line, especially when you know the slope and a point on the line. The formula is written as \( y - y_1 = m(x - x_1) \):

• \( m \) is the slope.
• \( (x_1, y_1) \) is a known point on the line.

This form is particularly handy for constructing an equation when you have specific points. Consider we have the slope 3 and a point \((1, 0)\). The point-slope form would be:

• \( y - 0 = 3(x - 1) \)
• \( y = 3x - 3 \)

Even though our example led to the slope-intercept form, point-slope is often your first step in forming a tailored equation. It's kind of a bridge between knowing specific line points and the easier slope-intercept formula.