Problem 33 - Thomas Calculus

Question

In Exercises \(31-34,\) find the line's \(x\) - and \(y\) -intercepts and use this information to graph the line. $$ \sqrt{2} x-\sqrt{3} y=\sqrt{6} $$

Step-by-Step Solution

Answer

The x-intercept is \((\sqrt{3}, 0)\) and the y-intercept is \((0, -\sqrt{2})\).

Step 1: Identify the Line Equation

The given equation of the line is \( \sqrt{2} x - \sqrt{3} y = \sqrt{6} \). We will use this equation to find both the \( x \)-intercept and the \( y \)-intercept.

Step 2: Find the x-intercept

To find the \( x \)-intercept, set \( y = 0 \) in the equation and solve for \( x \): \[\sqrt{2} x - \sqrt{3} \cdot 0 = \sqrt{6} \\sqrt{2} x = \sqrt{6} \x = \frac{\sqrt{6}}{\sqrt{2}} \\]To simplify \( \frac{\sqrt{6}}{\sqrt{2}} \), multiply both the numerator and the denominator by \( \sqrt{2} \):\[x = \frac{\sqrt{6} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{12}}{2} = \frac{2\sqrt{3}}{2} = \sqrt{3}\]Thus, the \( x \)-intercept is \( (\sqrt{3}, 0) \).

Step 3: Find the y-intercept

To find the \( y \)-intercept, set \( x = 0 \) in the equation and solve for \( y \): \[\sqrt{2} \cdot 0 - \sqrt{3} y = \sqrt{6} \- \sqrt{3} y = \sqrt{6} \y = -\frac{\sqrt{6}}{\sqrt{3}} \\]To simplify \( -\frac{\sqrt{6}}{\sqrt{3}} \), multiply both the numerator and the denominator by \( \sqrt{3} \):\[y = - \frac{\sqrt{6} \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = - \frac{\sqrt{18}}{3} = -\frac{3\sqrt{2}}{3} = -\sqrt{2}\]Thus, the \( y \)-intercept is \( (0, -\sqrt{2}) \).

Step 4: Graph the Line

Use the intercepts found to graph the line. Plot the \( x \)-intercept \( (\sqrt{3}, 0) \) and the \( y \)-intercept \( (0, -\sqrt{2}) \) on a coordinate plane. Draw a straight line through these points, extending in both directions.

Key Concepts

x-intercepts

The \( x \)-intercept of a line is the point where the line crosses the \( x \)-axis on a graph. To find the \( x \)-intercept, you set \( y \) equal to zero in the line equation and solve for \( x \).
In the equation \( \sqrt{2}x - \sqrt{3}y = \sqrt{6} \), we find the \( x \)-intercept by substituting \( y = 0 \). This simplifies the equation to \( \sqrt{2}x = \sqrt{6} \).
By solving, we find \( x = \frac{\sqrt{6}}{\sqrt{2}} \), which simplifies to \( \sqrt{3} \). The \( x \)-intercept is hence \((\sqrt{3}, 0)\).

• Set \( y = 0 \) in the equation.
• Solve the equation for \( x \).
• The solution gives the \( x \)-coordinate of the intercept.

Finding the \( x \)-intercept helps in graphing as it provides a definite point through which the line passes.

y-intercepts

The \( y \)-intercept of a line is the point where the line crosses the \( y \)-axis on a graph. To discover it, set \( x \) equal to zero and solve the resulting equation for \( y \). This removes any \( x \) dependency, making \( y \) isolated in the equation.
For the equation \( \sqrt{2}x - \sqrt{3}y = \sqrt{6} \), we set \( x = 0 \), transforming our equation to \( -\sqrt{3}y = \sqrt{6} \). Through solving, \( y = -\frac{\sqrt{6}}{\sqrt{3}} \) simplifies to \( -\sqrt{2} \). Thus, the \( y \)-intercept becomes \((0, -\sqrt{2})\).

• Set \( x = 0 \) in the line equation.
• Solve for \( y \).
• The resulting value is the \( y \)-coordinate of the intercept.

The \( y \)-intercept provides another crucial reference point for drawing the line on the coordinate plane.

coordinate plane

The coordinate plane is a two-dimensional space where each point is defined by an \( x \)-coordinate and a \( y \)-coordinate. It is a crucial concept in graphing lines and understanding linear equations.
The plane is divided into four quadrants by the \( x \)-axis, which runs horizontally, and the \( y \)-axis, which runs vertically.

• Quadrant I: Positive \( x \) and \( y \) coordinates.
• Quadrant II: Negative \( x \) and positive \( y \) coordinates.
• Quadrant III: Negative \( x \) and \( y \) coordinates.
• Quadrant IV: Positive \( x \) and negative \( y \) coordinates.

Understanding the axes and quadrants aids in accurately plotting points like \( x \)-intercepts and \( y \)-intercepts found from the equation. Once both intercepts are located on the coordinate plane, a line can be graphically represented by drawing a line through these points.

line equation

A line equation defines a straight line on the coordinate plane mathematically. It can be expressed in various forms, such as the standard form, slope-intercept form, or point-slope form.
In this problem, the equation \( \sqrt{2}x - \sqrt{3}y = \sqrt{6} \) is in the standard form \( Ax + By = C \). Here, each part represents:

• \( A \): The coefficient of \( x \).
• \( B \): The coefficient of \( y \).
• \( C \): The constant term on the right side.

The role of a line equation is significant in describing all the points lying on the line. Any equation can be manipulated to reveal different line characteristics, such as its slope, intercepts, or angle. It serves as the primary tool in identifying specific intercepts which are vital for graphing the line on the coordinate plane.