Problem 70 - Precalculus Essentials

Question

Use intercepts to graph equation. $$3 x+5 y+15=0$$

Step-by-Step Solution

Answer

The x-intercept of the equation is -5 and the y-intercept is -3. Plot these points and draw a line through them to graph the equation.

Step 1: Find the x-intercept

Set the y variable to 0, which gives us \n\(3x + 5(0) + 15 = 0\)\nThis can then be simplified to \n\(3x + 15 = 0\)\nSolve the equation for x to get the x-intercept:\n\(3x = -15\)\n\(x = -5 \) So, the x-intercept is -5

Step 2: Find the y-intercept

Set the x variable to 0, which gives us \n\(3(0) + 5y + 15 = 0\)\nThis can then be simplified to \n\(5y + 15 = 0\)\nSolve the equation for y to get the y-intercept: \n\(5y = -15\)\n\(y = -3\). So, the y-intercept is -3

Step 3: Graphing the Equation

Plot the x-coordinate at (-5,0) and the y-coordinate at (0,-3) on the graph. The line passing through these points is the graph of the equation.

Key Concepts

Understanding the x-intercept

To find the x-intercept of a linear equation like \(3x + 5y + 15 = 0\), you need to determine where the line crosses the x-axis. At this point, the y-coordinate is always 0 because the line doesn't rise or fall at the crossing. Therefore, you substitute \(y = 0\) into the equation, which gives:

• \(3x + 5(0) + 15 = 0\)
• Simplify to \(3x + 15 = 0\)
• Solving for \(x\), subtract 15 to get \(3x = -15\)
• Finally, divide by 3 to isolate \(x\), which yields \(x = -5\).

Thus, the x-intercept is the point \((-5, 0)\). Knowing how to find the x-intercept helps in understanding where the line will begin on the horizontal axis.

Finding the y-intercept

The y-intercept is where the line crosses the y-axis. At this point, \(x\) is always zero since the line hasn't moved from the y-axis horizontally. To find it, set \(x = 0\) in the equation \(3x + 5y + 15 = 0\):

• \(3(0) + 5y + 15 = 0\)
• Simplifies to \(5y + 15 = 0\)
• Subtract 15 to obtain \(5y = -15\)
• Dividing by 5 gets \(y = -3\).

So, the y-intercept is the point \((0, -3)\). Calculating the y-intercept is crucial as it shows where the line will pass through the vertical axis, giving a second anchor point for drawing the line.

Basics of solving equations

Solving equations like \(3x + 5y + 15 = 0\) involves isolating the variable you are interested in while keeping the equality balanced. For x-intercepts and y-intercepts, this means substituting values that simplify the equation.

• Set variable to zero (e.g., \(y = 0\) for x-intercept or \(x = 0\) for y-intercept).
• Simplify remaining terms to isolate the variable in question.
• Perform arithmetic operations such as addition/subtraction and multiplication/division.

This method ensures that you obtain the necessary intercept points without altering the balance of the equation. Understanding these steps enhances your ability to solve any linear equation effectively.

Graphing techniques for equations

Graphing linear equations using intercepts can be simplified with a few clear steps. Once you have the intercepts, you can easily plot the line:

• First, identify the intercepts. For our equation, these are \((-5, 0)\) and \((0, -3)\).
• On a coordinate graph, locate and mark these points.
• Draw a straight line through the two points. This line extends infinitely in both directions along the plane.

Understanding this technique is crucial as it uses minimal points to effectively chart the line's course, representing an infinite set of solutions. This visual representation helps in grasping the behavior of linear equations quickly and clarifies their solutions.