Problem 9 - Precalculus

Question

The graph of each equation is a straight line. Graph the equation after finding the \(x\)-and the \(y\) -intercepts. (since you are given that the graph is a line, you need only plot two points before drawing the line.) $$3 x+4 y=12$$

Step-by-Step Solution

Answer

The x-intercept is 4 and the y-intercept is 3. Plot (4,0) and (0,3) to draw the line.

Step 1: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation. The equation becomes \( 3x + 4(0) = 12 \). Solve for \( x \): \( 3x = 12 \). Divide both sides by 3 to get \( x = 4 \). So, the x-intercept is \( (4, 0) \).

Step 2: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation. The equation becomes \( 3(0) + 4y = 12 \). Simplify to get \( 4y = 12 \). Divide both sides by 4 to get \( y = 3 \). Thus, the y-intercept is \( (0, 3) \).

Step 3: Plot the points

Plot the x-intercept \((4, 0)\) and the y-intercept \( (0, 3) \) on a graph. These two points will allow you to draw the line of the equation.

Step 4: Draw the line

Use a ruler to draw a straight line through the points \((4, 0)\) and \((0, 3)\). This line represents the graph of the equation \(3x + 4y = 12\).

Key Concepts

X-intercept and Y-intercept

Understanding the x-intercept and y-intercept is crucial when learning how to graph linear equations. The x-intercept is the point where the line crosses the x-axis. To find this, you set the value of y to 0 and solve for x. Conversely, the y-intercept is where the line crosses the y-axis. Here, you'll set x to 0 and solve for y.

Finding these intercepts is often the first step in graphing a straight line since they provide exact coordinates that define the position of the line on the graph. For example, when given the equation \(3x + 4y = 12\), setting \(y = 0\) gives us the x-intercept, and setting \(x = 0\) gives us the y-intercept. Doing these calculations helps you to quickly identify the intercept points, such as \((4, 0)\) and \((0, 3)\) in this case.

These intercepts are like the anchors of your line, helping the line stay straight and true on your graph. Once these points are identified, plotting them is the next step in creating the visual representation of the equation.

Plotting Points

Plotting points on a graph is the essential step to transform numbers and equations into visual representations. To plot points effectively, you must understand that each point consists of an x-coordinate and a y-coordinate that we write in the form \(x, y\). These coordinates tell you exactly where on the graph to place your points.

In our example with the equation \(3x + 4y = 12\), we've calculated the intercepts as \((4, 0)\) and \((0, 3)\). Plotting these involves finding the point on the x-axis for \((4, 0)\), since y is 0 here, and then finding the point on the y-axis for \((0, 3)\), since x is 0 at this point. By accurately plotting these points, you set the foundation needed to draw the line representing the equation.

• Find the x-value on the horizontal axis and check the position.
• Check the y-value on the vertical axis and mark your point.
• Ensure each plotted point is correct to maintain accurate graphical representation.

These simple steps will help convert numerical solutions into visual graphs, which many find more intuitive and easier to understand.

Straight Line Equations

Straight line equations, also known as linear equations, are equations that make a straight line when they are graphed. These can be expressed in standard form as \(Ax + By = C\). The solution to these equations is plotted as a line in a two-dimensional space giving a straightforward relationship between the x and y variables.

Linear equations are significant in algebra because they represent constant, unchanging relationships. To graph a straight line equation like \(3x + 4y = 12\), you'll only need two points, the x-intercept and the y-intercept, to establish a line because these equations have a constant slope.

By drawing the line through the plotted points \((4, 0)\) and \((0, 3)\), you extend it across the graph. Using a ruler ensures the line remains straight, reflecting the uniform nature of these equations. This line visually represents all the solutions of the equation, showing how each x relates to a specific y.

Understanding the characteristics of straight line equations allows you to see broader patterns represented by these equations and apply this understanding to more complex problems.