Question

Find the x-intercept and the y-intercept of the graph of the equation. $$x-6 y=4$$

Short answer

The x-intercept is 4 and the y-intercept is -2/3.

Step-by-step solution

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). So the equation becomes \(x - 6*0 = 4\), therefore \(x = 4\). So the x-intercept is 4.

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). The equation becomes \(0 -6y = 4\), therefore, \(y = -4/6\). So the y-intercept is -2/3.

Key concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can always be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

For the equation x - 6y = 4, it's a linear equation with a = 1, b = -6, and c = 4. Understanding the structure of a linear equation is crucial as it guides how to find x-intercepts and y-intercepts, which are the points where the graph of the equation crosses the x-axis and y-axis, respectively.

Solving Equations

To solve an equation means to find the value of the variable that makes the equation true. When solving for the x-intercept in our example, we set y = 0 because the x-intercept will occur where the graph crosses the x-axis, and at any point on the x-axis, the y-value is zero. Similarly, solving for the y-intercept involves setting x = 0, as any point on the y-axis has an x-value of zero.

By substituting these values into the linear equation, we simplify and solve for the remaining variable, resulting in the intercepts: the x-intercept at x = 4 and the y-intercept at y = -2/3. It's crucial to isolate the variable on one side of the equation to efficiently find its value.

Graphing Linear Equations

The process of graphing linear equations involves plotting points on a coordinate plane that represent solutions to the equation. To graph the equation x - 6y = 4, we need two key points: where the line crosses the x-axis (the x-intercept) and where it crosses the y-axis (the y-intercept).

The x-intercept at (4, 0) and the y-intercept at (0, -2/3) are the points where the line touches each axis. Plotting these points and drawing a line through them will give you the graph of the equation. Remember, a straight line only needs two points to be fully defined on a graph, and intercepts are the most commonly used points for this purpose as they are relatively easy to find.