Problem 70 - College Algebra

Question

Use intercepts to graph each equation. \(3 x+5 y+15-0\)

Step-by-Step Solution

Answer

The x-intercept is (-5,0) and the y-intercept is (0, -3). The graph is a straight line passing through these intercepts.

Step 1: Find the x-intercept

The x-intercept is found by setting y=0 and then solving for x. In the given equation, \(3 x+5 (0)+15=0\), simplifies to \(3x = -15\). So, \(x = -15 / 3\) which is \(x = -5\). The x-intercept is thus, (-5,0).

Step 2: Find the y-intercept

The y-intercept is found by setting x=0 and then solving for y. In the given equation, \(3 (0)+5y+15=0\), simplifies to \(5y = -15\). So, \(y = -15 / 5\) which is \(y = -3\). The y-intercept is thus, (0,-3).

Step 3: Graph the line

Plot these intercepts on the graph and then draw a line that passes through these two points. This line represents the solution of the given linear equation.

Key Concepts

Understanding the X-Intercept

The x-intercept of a linear equation is the point where the line crosses the x-axis on a graph. At this point, the value of y is always 0. To find the x-intercept of an equation, you set y to 0 and solve for x.

For example, in the equation given in the exercise, we have:

• Start with the equation: \(3x + 5y + 15 = 0\).
• Set \(y = 0\): \(3x + 5(0) + 15 = 0\).
• This simplifies to \(3x + 15 = 0\).
• Solving for x gives \(3x = -15\).
• Divide both sides by 3 to find \(x = -5\).

So, the x-intercept is (-5, 0). This means the line will cross the x-axis at the point (-5, 0). When graphing, always remember that finding the x-intercept involves making y zero.

Understanding the Y-Intercept

The y-intercept of a linear equation is the point where the line crosses the y-axis. At the y-intercept, the value of x is always 0. To find the y-intercept, you set x to 0 and then solve for y.

Here's how it works using the equation from the exercise as an example:

• Start with the equation: \(3x + 5y + 15 = 0\).
• Set \(x = 0\): \(3(0) + 5y + 15 = 0\).
• This simplifies to \(5y + 15 = 0\).
• Solve for y: \(5y = -15\).
• Divide both sides by 5 to find \(y = -3\).

The y-intercept is (0, -3). This means that the line crosses the y-axis at the point (0, -3). On the graph, this is where the line will reach the vertical y-axis when x is zero.

Exploring Linear Equations

Linear equations represent straight lines on a graph, and they come in the form \(Ax + By + C = 0\). These equations are about finding a relationship between two variables, typically x and y.

To graph a linear equation, you need at least two points to draw the line. This is where intercepts become very useful:

• The x-intercept gives one point on the graph where y is zero.
• The y-intercept provides another point where x is zero.

Once you find these intercepts, you can graph them as points on the Cartesian plane. Connect these points with a straight line, and you will have the graph of the linear equation.

Linear equations are foundational in algebra as they describe direct relationships between variables. This means when one variable changes, the other changes at a consistent rate. Understanding how to work with linear equations is crucial for solving many real-world problems, such as calculating distances, budgeting expenses, and predicting trends.