Problem 70 - College Algebra Essentials

Question

Use intercepts to graph the each equation. $$3 x+5 y+15-0$$

Step-by-Step Solution

Answer

The x-intercept of the equation is (-5,0), the y-intercept is (0,-3). Plotting these two points and drawing a line through them provides the graph of the equation.

Step 1: Identify the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. Using the equation \(3x + 5 * 0 + 15 = 0\), simplifies to \(3x + 15 = 0\). Solving for x results in \(x = -15/3 = -5\). So, the x-intercept is at (-5,0).

Step 2: Identify the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Using the equation \(3 * 0 + 5y + 15 = 0\), simplifies to \(5y + 15 = 0\). Solving for y, results in \(y = -15/5 = -3\). So, the y-intercept is at (0,-3).

Step 3: Graph the line

Plot the identified intercepts (-5,0) and (0,-3) on a graph. Draw a line passing through these two points. The line represents the graph of the equation \(3x + 5y + 15 = 0\).

Key Concepts

x-intercept

The x-intercept of a linear equation is the point where the graph of the equation crosses the x-axis. This means that at the x-intercept, the value of y is always 0.
To find the x-intercept, you replace y with 0 in the equation and solve for x. For example, if we have the equation \(3x + 5y + 15 = 0\), to find the x-intercept, we set \(y = 0\) which simplifies the equation to \(3x + 15 = 0\). Solving this equation leads to the calculation \(x = -\frac{15}{3} = -5\).
Thus, the x-intercept is at the point \((-5, 0)\). Knowing this point is crucial as it helps you graph the line by giving you a specific location where the line interacts with the x-axis.

• Remember: At the x-intercept, \(y = 0\)
• Solve for x by substituting \(y = 0\) into your equation
• This process locates the point \((x, 0)\) on the x-axis

y-intercept

The y-intercept is the point where the graph of a linear equation crosses the y-axis, where the value of x is always 0.
To find the y-intercept, set x to 0 and solve for y. Using the same example equation, \(3x + 5y + 15 = 0\), setting \(x = 0\) leads to the equation \(5y + 15 = 0\). Solving for y results in \(y = -\frac{15}{5} = -3\).
This gives us the y-intercept at \((0, -3)\). The y-intercept provides a clear point where the line intersects the y-axis, which is essential for graphing the equation as it gives an additional accurate point.

• At the y-intercept, \(x = 0\)
• Solve the equation by substituting \(x = 0\)
• Find the point \((0, y)\) on the y-axis

coordinate plane

The coordinate plane is a two-dimensional surface that allows us to locate points using pairs of numbers, known as coordinates. It consists of a horizontal axis, typically called the x-axis, and a vertical axis, known as the y-axis.
Each point on the plane is described by an ordered pair \((x, y)\), where x indicates the horizontal position and y the vertical position. Understanding the coordinate plane is fundamental for graphing any linear equation or understanding the geometrical representation of algebraic expressions.

• The x-axis runs horizontally, left to right
• The y-axis runs vertically, top to bottom
• The point \((0, 0)\) is known as the origin

Graphing linear equations involves locating and plotting points accurately on this plane, followed by drawing a line through them. As we plot the intercepts like \((-5, 0)\) and \((0, -3)\), we use the coordinate plane to help align and extend the line correctly.

plotting points

Plotting points involves marking the location of coordinates on the coordinate plane. This is an essential initial step when graphing a line from a linear equation. By identifying key points such as the x-intercept and y-intercept, you lay down the foundation for drawing the linear graph accurately.
When plotting points, it's crucial to understand their coordinates. For instance, the points \((-5, 0)\) and \((0, -3)\) tell you exactly where to place them on the x-axis and y-axis respectively. Each coordinate pair gives a specific position on the grid where x and y meet.

• Ensure accuracy by checking both x and y values
• Mark each point carefully on the plane
• Use several points for clear and precise graph representation

Once plotted, these points guide you to draw the correct line that represents the equation, ensuring that you visualize the linear relationship effectively.