Problem 72 - College Algebra

Question

Use intercepts to graph each equation. \(6 x-3 y+15-0\)

Step-by-Step Solution

Answer

The x-intercept is (-2.5,0) and the y-intercept is (0,5). The line passing through these two points is the graph of the given equation.

Step 1: Find the X-intercept

To find the x-intercept, we replace y in our equation with zero. Our equation becomes \(6x + 15 = 0\). After rearranging it, we get \(x = -15/6 = -5/2\). So the x-intercept is \(-5/2 or -2.5\).

Step 2: Find the Y-intercept

To find the y-intercept, replace x in our equation with zero. The equation then changes to \(-3y + 15 = 0\). After solving for y, we get \(y = 15 / 3 = 5\). Therefore, the y-intercept is \(5\).

Step 3. Graphing the intercepts

Now we graph the points we found in steps 1 and 2. To do this, on the x-axis locate -2.5 (the x-intercept) and mark it, then on the y-axis, locate 5 (the y-intercept) and mark it. Then draw a line passing through these two points. This line is the graph of the given equation.

Key Concepts

x-intercept

When graphing linear equations, one of the first steps involves finding the x-intercept. The x-intercept is where the graph of the equation crosses the x-axis. At this point, the value of y is always zero. To find the x-intercept for the equation provided in the exercise, set y to zero and solve for x.
Consider the equation:

• Start with: \(6x - 3y + 15 = 0\)
• Set \(y = 0\) to find the x-intercept.
• This simplifies to \(6x + 15 = 0\).
• Solving for x, you get \(x = -\frac{15}{6}\), which reduces to \(-2.5\).

The x-intercept is the point \((-2.5, 0)\). This point indicates where the line will touch or cross the x-axis.

y-intercept

Another important step in graphing a linear equation is finding the y-intercept. The y-intercept is where the graph crosses the y-axis. At this point, the value of x is always zero. To determine the y-intercept, substitute x with zero in the equation and solve for y.
Using the same equation, let's find the y-intercept:

• Start with: \(6x - 3y + 15 = 0\)
• Set \(x = 0\) to find the y-intercept.
• The equation becomes \(-3y + 15 = 0\).
• Solving for y, you get \(y = \frac{15}{3} = 5\).

The y-intercept is the point \((0, 5)\). This is where the line will intersect the y-axis.

plotting points

After finding both the x-intercept and y-intercept, plotting these points on a coordinate plane is the next step to graphing the linear equation. These intercepts provide a clear visual starting and ending point for the line representing the equation.
To accurately plot the intercepts:

• Locate the x-intercept \((-2.5, 0)\) on the x-axis.
• Mark this point clearly on your graph.
• Then, find the y-intercept \((0, 5)\) on the y-axis.
• Mark this point distinctively as well.

Once both points are plotted, draw a straight line through them. The line should extend through the plane, accurately representing the equation \(6x - 3y + 15 = 0\). This visual representation can help you understand the relationship between x and y in the equation.

solving equations

Solving equations is a fundamental skill in graphing and understanding linear equations. The process involves manipulating the equation to find specific points, like intercepts, that make graphing possible.
Here's how you solve for these intercepts:

• For the x-intercept, substitute \(y = 0\) into the equation and solve for x.
• For the y-intercept, substitute \(x = 0\) into the equation and solve for y.

The key steps are:

• Rearrange the equation as needed to isolate the variable of interest.
• Use algebraic techniques, such as adding, subtracting, multiplying, or dividing both sides of the equation.

These methods help facilitate the process of finding exact values for intercepts, essential for accurate graphing. Understanding how to solve equations and use intercepts empowers you to portray complex relationships visually.