Problem 69 - College Algebra Essentials

Question

Use intercepts to graph the each equation. $$2 x+3 y+6-0$$

Step-by-Step Solution

Answer

The x-intercept is \(-3\) and the y-intercept is \(-2\). Plot these two points on a graph and draw a line through them to represent the equation.

Step 1: Rearrange the equation

Rearrange the equation \(2x + 3y + 6 = 0\) to standard form.

Step 2: Find the x-intercept

Set \(y = 0\) in the equation and solve for \(x\). This gives the x-intercept.

Step 3: Find the y-intercept

Set \(x = 0\) in the equation and solve for \(y\). This gives the y-intercept.

Step 4: Plot the intercepts and graph

Plot the x and y intercepts on a graph. Draw a line through these two points to represent the equation.

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 be represented graphically on a coordinate plane, where they appear as straight lines. These lines show the relationship between two variables, often denoted by x and y. A unique property of linear equations is that they have a constant rate of change, also known as the slope.

X-Intercept

The x-intercept of a graph is the point where the line crosses the x-axis. To find this intercept for a linear equation, you set the y value to zero and solve for x. This is because the x-axis corresponds to y=0, and any point on this axis will not have any y component. It is a key element to understanding the graph structure and helps with drawing the line on the coordinate plane.

Y-Intercept

Conversely, the y-intercept is where the line crosses the y-axis. To calculate the y-intercept, you assign the value of zero to x and solve for y. It indicates the point where the x-axis is at x=0. The y-intercept provides valuable information about the line, such as where it starts on the graph if you were to follow it from left to right.

Standard Form of a Linear Equation

The standard form of a linear equation is written as Ax + By + C = 0, where A, B, and C are integers, and A and B are not both zero. Converting a linear equation to its standard form can often make finding intercepts more straightforward. When an equation is in standard form, it's very easy to identify and use the coefficients directly to find the x and y intercepts.

Plotting Intercepts on a Graph

After finding the x and y intercepts, the next step is plotting intercepts on a graph. You mark the x-intercept on the x-axis and the y-intercept on the y-axis. These two points are all you need to draw a straight line representing the equation since a line in a two-dimensional plane is uniquely defined by two points. Joining the intercepts creates a visual representation that helps in understanding the equation's solutions and behavior.