Question

Find the value of \(c\) for which the line \(3 x+c y=5\) (a) passes through the point \((3,1)\); (b) is parallel to the \(y\) -axis; (c) is parallel to the line \(2 x+y=-1\); (d) has equal \(x\) - and \(y\) -intercepts; (e) is perpendicular to the line \(y-2=3(x+3)\).

Short answer

(a) \(c = -4\); (b) \(c = 0\); (c) \(c = \frac{3}{2}\); (d) \(c = -3\); (e) \(c = 9\).

Step-by-step solution

Check for passing through a specific point

If the line passes through the point \((3,1)\), we substitute these values into the equation \(3x + cy = 5\). Substituting, we get:\[3(3) + c(1) = 5\]\[9 + c = 5\]Solving for \(c\), we find that:\[c = 5 - 9 = -4\]

Determine line parallel to the y-axis

For a line to be parallel to the \(y\)-axis, its equation must have the form \(x = a\), meaning the coefficient of \(y\) should be zero in the equation \(3x + cy = 5\). Thus, \(c = 0\).

Determine line parallel to another given line

If two lines are parallel, their slopes must be equal. The given line \(2x + y = -1\) can be rewritten in slope-intercept form as \(y = -2x - 1\) with a slope of \(-2\). The given line \(3x + cy = 5\) can be rewritten as \(y = \frac{-3}{c}x + \frac{5}{c}\). Setting the slope \(\frac{-3}{c} = -2\), we solve:\[-3 = -2c\]\[c = \frac{3}{2}\]

Find equal x- and y-intercepts

For a line with equal \(x\)- and \(y\)-intercepts, set \(\frac{5}{3}\) (x-intercept) equal to \(-\frac{5}{c}\) (y-intercept):\[\frac{5}{3} = -\frac{5}{c}\]By cross-multiplying, we solve for \(c\):\[5c = -15\]\[c = -3\]

Determine line perpendicular to another given line

A line is perpendicular to another if the product of their slopes is \(-1\). From the line equation \(y - 2 = 3(x + 3)\), we find its slope to be \(3\). Thus, the perpendicular slope should be \(-\frac{1}{3}\). For our line \(3x + cy = 5\) in slope-intercept form, the slope is \(-\frac{3}{c}\). We solve:\[-\frac{3}{c} \times 3 = -1\]\[-\frac{9}{c} = -1\]\[c = 9\]

Key concepts

Linear equations

Linear equations form the cornerstone of many mathematical applications. They consist of variables which are typically combined using addition or subtraction, set equal to a constant. These equations graph as straight lines on a coordinate plane. In the form \(ax + by = c\), \(a\), \(b\), and \(c\) are constants, while \(x\) and \(y\) are the variables. Understanding these equations is essential, as they allow us to model relationships between quantities and solve problems involving predictions or optimizations. They also serve as a stepping stone to more complex mathematics, making it crucial to grasp their properties and solutions.
To solve a linear equation, our goal often involves isolating one variable. This goal enables us to graph the equation or use it in further calculations. Linear equations are ubiquitous in everyday life, appearing in situations from simple budgets to complex scientific models.

Slope-intercept form

The slope-intercept form of a linear equation is one of the most used forms, often written as \(y = mx + b\). Here, \(m\) represents the slope of the line, indicating how steep the line is. The \(b\) is the y-intercept, which marks the point where the line crosses the y-axis.

• *Slope* tells us how much the y-value changes with a change in x-value.
• *Y-intercept* provides the line's starting point on the y-axis.

This form makes it easy to graph the line and understand its behavior without complex calculations.
For example, to check if two lines are parallel in the exercise, we compare their slopes. If they have the same slope, they are parallel. If you encountered an equation such as \(3x + cy = 5\), you would reformat it to visualize the slope and intercept: \(y = \frac{-3}{c}x + \frac{5}{c}\). Adjusting for \(c\) directly influences these parameters, tailoring the line's position and inclination.

Parallel and perpendicular lines

In coordinate geometry, parallel and perpendicular lines have specific relationships. Lines are parallel when they have the same slope but different y-intercepts. This means they never meet, just like railroad tracks. On the other hand, perpendicular lines intersect at a right angle (90 degrees).

• *Parallel lines* maintain equal slope values, seen in the solution where equations like \(y = -2x - 1\) are compared.
• *Perpendicular lines* feature slopes that are negative reciprocals of each other, like how \(3x + cy = 5\) becomes perpendicular when transformed to match the negative reciprocal \(-\frac{1}{3}\).

Understanding these properties allows not only for solving problems involving line arrangements but also for recognizing and predicting patterns in graphs, visuals, and even real-world applications.

Coordinate geometry

Coordinate geometry allows us to describe geometric shapes and their properties using numbers and equations. It blends algebra with geometry through the use of coordinates to represent points on a plane. Each point is identified by an \(x\) and \(y\) pair, corresponding to horizontal and vertical position.
Key aspects are identifying the equation of lines and the relationships between them. Such was the case when determining if a line passes through a specific point, is parallel, or perpendicular to another line.
For example, the exercise demonstrates decision-making using equations involving the line \(3x + cy = 5\). By substituting coordinates or observing slopes and intercepts, we gain insights into the geometric configurations on the plane.
This approach extends to a variety of applications like mapping, architecture, and animations, providing an invaluable tool for visually representing and manipulating space.