Question

Write the equation in slope-intercept form. Then graph the equation. $$ 2 x-3 y-6=0 $$

Short answer

The slope-intercept form of the equation is \( y = 2/3x - 2 \). The graph of this equation starts at the point (0, -2) and rises 2 units and runs 3 units to the right.

Step-by-step solution

Convert to Slope-Intercept Form

Begin by rearranging the equation in the form \( y = mx + b \). To do this, solve the equation \(2x-3y-6=0 \) for y. \Begin by subtracting 2x from both sides to isolate the y term: \( -3y = -2x + 6 \). Then, divide all the terms by -3 to solve for y: \(y = 2/3x - 2\)

Identify the Slope and Y-intercept

In the equation \( y = 2/3x - 2 \), the slope (m) is 2/3 and the y-intercept (b) is -2. The slope indicates that for every increase of 1 in x, y increases by 2/3, and the y-intercept is the point on the y-axis where the line crosses, which is (0, -2).

Graph the Equation

To graph the equation \(y = 2/3x - 2\), start by plotting the y-intercept (-2) on the y-axis. From this point, use the slope to find the next point. Since the slope is 2/3, rise 2 units and run 3 units to the right. This provides the second point (3, 0). Draw a line connecting these points, and extend it. This line represents the graph of the equation.

Key concepts

Linear Equations

Linear equations are fundamental in algebra and represent straight lines when graphed on a coordinate plane. These equations can be recognized by their standard form, which appears as either ax + by = c, where a, b, and c are constants, or in slope-intercept form, which is y = mx + b. Here, m represents the slope of the line and b is the y-intercept, indicating where the line crosses the y-axis. These equations show a direct proportional relationship between the variables x and y.

When you encounter a linear equation like 2x - 3y - 6 = 0, your first task is often to solve for y to get it into slope-intercept form. This enables you to easily understand the rate of change in the relationship between x and y and to quickly graph the line.

Graphing Linear Equations

Graphing linear equations involves plotting the relationship between two variables on an xy-coordinate plane. To graph a linear equation that's already in slope-intercept form, y = mx + b, you start at the y-intercept, the point where x equals zero. Next, you use the slope, 'm', which tells you how to move from one point on the line to another. The slope is a ratio that indicates how many units to 'rise' (move up or down) and 'run' (move left or right).

For the equation y = 2/3x - 2, you'll begin by plotting the y-intercept (0, -2) on the graph. Then, using the slope of 2/3, from the y-intercept, you move up 2 units and to the right 3 units to find the next point. By connecting these points with a straight line and extending it across the graph, you visualize the relationship dictated by the equation.

Slope of a Line

The slope of a line in a linear equation represents the steepness or the angle of the line when graphed. It's calculated as 'rise over run' or the change in y (the vertical axis) divided by the change in x (the horizontal axis). Significantly, the slope is the 'm' in the slope-intercept form, y = mx + b.

If a line has a slope of 2/3, it means that for every 3 units the line moves horizontally to the right (the run), it will move 2 units vertically upwards (the rise). A positive slope like this indicates an increasing line, while a negative slope represents a decreasing line. The greater the magnitude of the slope, the steeper the incline or decline of the line. A slope of zero indicates a horizontal line, and an undefined slope (when the denominator is zero) signifies a vertical line.

Y-Intercept

The y-intercept is a specific point on a graph where a line crosses the y-axis. It corresponds to the 'b' value in the slope-intercept equation y = mx + b. It's important because it provides a starting reference point for graphing the line and is the precise spot the line would touch the y-axis when the value of x is zero.

In the context of our example equation, y = 2/3x - 2, the y-intercept is -2. This means the line crosses the y-axis at the point (0, -2). The y-intercept is essential in graphing because it tells us where to begin plotting the line and helps visualize where the line would be positioned relative to the origin of the coordinate plane.