Problem 56 - Introductory Algebra for College Students

Question

Use intercepts to graph \(3 x-5 y=15.\) (Section 3.2, Example 4)

Step-by-Step Solution

Answer

The graph of the linear equation \(3x - 5y = 15\), passes through the points (5,0) (x-intercept) and (0,-3) (y-intercept).

Step 1: Find the x-intercept

The x-intercept can be found by setting y=0 and solving for x. Substituting y = 0 into the equation \(3x - 5y = 15\), the equation simplifies to \(3x = 15\). Solving this equation gives x = 5. So, the x-intercept is at point (5,0).

Step 2: Find the y-intercept

The y-intercept can be found by setting x=0 and solving for y. Substituting x = 0 into the equation \(3x - 5y = 15\), the equation simplifies to \(-5y = 15\). Solving this equation gives y = -3. So, the y-intercept is at point (0,-3).

Step 3: Graphing the linear equation

Now the equation can be graphed using the x-intercept at the point (5,0) and the y-intercept at the point (0,-3). These points are marked on the plane, and a straight line is drawn through them, which graphs the linear equation.

Key Concepts

Understanding the X-Intercept

The x-intercept is a key component when graphing linear equations. It is where the graph of an equation crosses the x-axis. This means the y-coordinate at this point is zero because it's located exactly on the horizontal axis.

To find the x-intercept from an equation, you need to set the y value to zero and solve for x. Let's take the equation in the problem: \(3x - 5y = 15\). If we substitute \(y = 0\), the equation simplifies to \(3x = 15\). Solving for \(x\) gives you \(x = 5\). Hence, the x-intercept for this equation is at the point (5,0).

In summary:

• Set \(y = 0\).
• Solve the equation for \(x\).
• Find the x-intercept point, which is where the graph touches the x-axis.

Decoding the Y-Intercept

Just like the x-intercept, the y-intercept is crucial in understanding the behavior of linear equations. The y-intercept is where the graph crosses the y-axis. Here, the x-coordinate is zero because the point lies directly on the vertical axis.

To find the y-intercept, set the x value to zero in the original equation and solve for y. With the equation \(3x - 5y = 15\), if substitute \(x = 0\), it reduces to \(-5y = 15\). Solving for \(y\) results in \(y = -3\), placing the y-intercept at (0, -3).

Remember these steps:

• Set \(x = 0\).
• Solve the equation for \(y\).
• Identify the y-intercept, the point where the line crosses the y-axis.

Introduction to Algebraic Equations in Graphing

Algebraic equations form the backbone of many mathematical tasks including graphing. They show relationships between numbers and variables and can be used to describe curves and lines on a graph.

A linear equation, like the one in the exercise \(3x - 5y = 15\), represents a straight line when graphed. This form is important as it allows us to predict the line's behavior using intercepts. The slope-intercept form of a line is particularly useful: \(y = mx + b\). Here, \(m\) denotes the slope, and \(b\) represents the y-intercept.

Linear equations are easy to work with because:

• They involve constant rates of change, resulting in straight lines.
• Solving for one variable in terms of another helps in visualizing the graph.

Graphing linear equations by finding x and y intercepts provides a simple approach to visualize and better understand these basic algebraic concepts.