Problem 47 - Algebra for College Students

Question

Find the \(x\) - and \(y\) -intercepts for each line and use them to graph the line. $$4 x-3 y=12$$

Step-by-Step Solution

Answer

x-intercept is (3,0) and y-intercept is (0,-4)

Step 1: Find the x-intercept

To find the x-intercept, set y to 0 in the equation \[4x - 3y = 12\]So, replace y with 0:\[4x - 3(0) = 12\]This simplifies to \[4x = 12\].Solving for x, we get \[x = \frac{12}{4} = 3\].Therefore, the x-intercept is at the point (3, 0).

Step 2: Find the y-intercept

To find the y-intercept, set x to 0 in the equation \[4x - 3y = 12\]So, replace x with 0:\[4(0) - 3y = 12\]This simplifies to \[-3y = 12\].Solving for y, we get \[y = \frac{12}{-3} = -4\].Therefore, the y-intercept is at the point (0, -4).

Step 3: Plot the intercepts

Now, plot the x-intercept (3, 0) and the y-intercept (0, -4) on the coordinate plane.

Step 4: Draw the line

Draw a line through the points (3, 0) and (0, -4) to represent the equation \[4x - 3y = 12\].

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning they graph as straight lines. They usually appear in the form [a x + b y = c] where [a], [b], and [c] are constants. In your equation[4x - 3y = 12], [4] and [-3] are the coefficients of [x] and [y], while [12] is the constant term.

The goal is to make sure the equation balances. You'll often work by isolating one variable. For instance, solving for [x] or [y] helps when finding intercepts. The solutions (x, y) represent points on the coordinate plane.

• Key Feature: Linear equations make lines when plotted on a graph.
• Variables: Typically involve [x] and [y].
• Balance: The goal is to isolate variables to simplify the equation.
• Slope and Intercept: Slope ([m]) tells the tilt of the line, and intercept refers to where it crosses the axes.

Graphing Lines

Graphing lines involves plotting points on a coordinate plane based on an equation. Here, you found the xeq [3] intercept by setting [y]eq [0] and solving [4(3) - 3(0) = 12]. You also found the yeq [-4] intercept by setting [x]eq [0] and solving [4(0) - 3(-4) = 12].

Once you have these intercepts—essentially points [(3,0)] and [(0,-4)]—you can plot them on the plane. The line connecting these dots represents your equation. This is an effective way to visualize linear equations and see solutions.

• Plot Points: Start by plotting the intercepts.
• Draw Line: Connect the points to form a straight line.
• Equation Representation: The line is a visual representation of your equation.
• Verify: Points along this line satisfy your original equation.

Coordinate Plane

The coordinate plane is a two-dimensional surface formed by the x-axis (horizontal) and y-axis (vertical). It is used to plot points, which are written as ([x]eq, [y]eq) pairs. The x-intercept is where the line crosses the x-axis (y = 0), and the y-intercept is where it crosses the y-axis (x = 0). In your case, the intercepts were (3, 0) and (0, -4).

This system makes it easy to visualize equations and their solutions. By plotting points where the line intersects axes, you turn abstract equations into concrete, visual solutions.

• Axes: Horizontal (x-axis) and vertical (y-axis) lines.
• Intersections: Points where the line crosses axes are intercepts.
• Plotting Points: Use (x, y) coordinates to represent points.
• Graphing: Draw lines through plotted points for visualization.