Problem 11 - Introductory Algebra for College Students

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Do not graph the equation. \(2 x-3 y=15\)

Step-by-Step Solution

Answer

The \(x\)-intercept is \(7.5\), and the \(y\)-intercept is \(-5\).

Step 1: Find the x-intercept

To find the \(x\)-intercept, set \(y=0\) in the given equation. That leads to the equation \(2x - 3(0) = 15\), which simplifies to \(2x = 15\). Dividing both sides by 2 gives \(x = 15 / 2 = 7.5\). Thus, the \(x\)-intercept is \(7.5\).

Step 2: Find the y-intercept

Similarly, to find the \(y\)-intercept, set \(x=0\) in the given equation. That leads to the equation \(2(0) - 3y = 15\), which simplifies to \(-3y = 15\). Dividing both sides by \(-3\) gives \(y = 15 / -3 = -5\). Thus, the \(y\)-intercept is \(-5\).

Key Concepts

Linear Equations

A linear equation is the foundation of algebra and coordinate geometry, characterized by having the highest power of variables as one. The general form of a linear equation in two variables, like 'x' and 'y', is expressed as \( Ax + By = C \) where 'A', 'B', and 'C' are constants. These equations represent straight lines when graphed on a Cartesian plane.

When it comes to finding intercepts, it's valuable to remember that the x-intercept is the point where the line crosses the x-axis (hence, \( y = 0 \) ), and the y-intercept is where the line crosses the y-axis (where \( x = 0 \) ). In our example, \( 2x - 3y = 15 \) is a linear equation and by altering 'x' or 'y' to zero, we can easily find these intercepts.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, facilitates the link between geometry and algebra through a system of coordinates. It uses numerical values to represent geometric shapes and their properties. The most basic coordinate system is the two-dimensional Cartesian coordinate system, where each point is defined by an ordered pair of numbers \( (x, y) \).

In terms of linear equations and intercepts, the x-intercept is the coordinate where the line meets the x-axis and is always of the form \( (x, 0) \) while the y-intercept is where the line meets the y-axis, given as \( (0, y) \). These intercepts are crucial in sketching the graph of the equation and can reveal much about the equation's geometric properties.

Algebraic Methods

Algebraic methods refer to the various techniques used to manipulate and solve equations. One primary technique is substitution, which we used in finding the intercepts; it involved replacing a variable with a value (zero in this case) to solve for the other variable.

Other methods include rearranging the equation, factoring, and using the inverse operations. In our example, setting \( y = 0 \) or \( x = 0 \) simplifies the original equation to a single variable, thereby making it solvable. Algebra is immensely powerful in systematically solving problems that involve unknown quantities.

Graphical Interpretation

Graphical interpretation involves understanding and analyzing mathematical concepts through their visual representation. For a linear equation such as \( 2x - 3y = 15 \) plotting this on a graph yields a straight line. The intercept points—where the line crosses the axes—are pivotal in graph reading and outline the linear function's direction and slope.

The x-intercept grasped by setting \( y=0 \) and solving for 'x', informs us where the graph cuts the x-axis. Conversely, the y-intercept achieved by setting \( x=0 \) and solving for 'y', tells us where it cuts the y-axis. These points can be a significant initial step in sketching the full line on the graph.