Problem 58 - College Algebra with Modeling and Visualization

Question

Determine the \(x\) - and \(y\) -intercepts on the graph of the equation. Graph the equation. \(-3 x-5 y=15\)

Step-by-Step Solution

Answer

The x-intercept is (-5, 0) and the y-intercept is (0, -3).

Step 1: Understanding the Equation

The given equation to graph is a linear equation: \[-3x - 5y = 15\]Our task is to determine the x-intercept and y-intercept, and then graph the line.

Step 2: Finding the X-Intercept

To find the x-intercept, set \(y=0\) in the equation and solve for \(x\):\[-3x - 5(0) = 15\]\[-3x = 15\]Now, divide both sides by \(-3\):\[x = -5\]Therefore, the x-intercept is at the point \((-5, 0)\).

Step 3: Finding the Y-Intercept

To find the y-intercept, set \(x=0\) in the equation and solve for \(y\):\[-3(0) - 5y = 15\]\[-5y = 15\]Now, divide both sides by \(-5\):\[y = -3\]Therefore, the y-intercept is at the point \((0, -3)\).

Step 4: Graphing the Equation

Plot both intercepts on a coordinate plane. The x-intercept \((-5, 0)\) is on the x-axis, and the y-intercept \((0, -3)\) is on the y-axis. Draw a straight line through these two points, extending in both directions. This line represents the graph of the equation \(-3x - 5y = 15\).

Key Concepts

X-Intercept

An x-intercept of a linear equation is the point where the line crosses the x-axis. It is the value of \( x \) when \( y \) is zero, because at any point on the x-axis, the value of \( y \) is always zero. To find the x-intercept of the equation \(-3x - 5y = 15\), we substitute \( y = 0 \) into the equation and solve for \( x \).

Given,

• \(-3x - 5 \cdot 0 = 15\)
• \(-3x = 15\)
• Dividing both sides by \(-3\): \(x = -5\)

Therefore, the x-intercept is \((-5, 0)\). This means the line passes through the point \((-5, 0)\) on the graph. Finding intercepts is useful because it gives us clear points to plot, helping quickly sketch the line of a linear graph.

Y-Intercept

A y-intercept is where the line crosses the y-axis. At this point, the value of \( x \) is zero because the y-axis intercept is the line of \( x = 0 \). To determine the y-intercept for the equation \(-3x - 5y = 15\), we set \( x = 0 \) and solve for \( y \).

Let's simplify the equation:

• \(-3 \cdot 0 - 5y = 15\)
• \(-5y = 15\)
• By dividing both sides by \(-5\): \(y = -3\)

Thus, the y-intercept is at \((0, -3)\). This point shows where the line crosses the y-axis. Knowing the intercepts like this makes it straightforward to plot these crucial points for graphing the line.

Graphing Linear Equations

Graphing linear equations is like connecting dots on a plot. Once you find the x- and y-intercepts, you can easily place these points on a coordinate system. For the equation \(-3x - 5y = 15\), we found the intercepts as \((-5, 0)\) and \((0, -3)\). These intercepts are critical because they give us two precise points where the line crosses the x-axis and y-axis.

To graph the equation properly:

• Plot the x-intercept \((-5, 0)\) on the x-axis.
• Plot the y-intercept \((0, -3)\) on the y-axis.
• Draw a straight line through these two points. Extend the line in both directions.

This line is the graph of the equation \(-3x - 5y = 15\). Linear equations will always form straight lines, which represent the infinitely many solutions to the equation. The intercepts are significant because they provide easy reference points to start your graph.