Question

In Problems \(99-108\), (a) find the intercepts of the graph of each equation and (b) graph the equation. $$ 2 x+3 y=6 $$

Short answer

The intercepts are (3, 0) and (0, 2). Plot these points and draw the line through them.

Step-by-step solution

- Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x: \[ 2x + 3(0) = 6 \] Simplify to get: \[ 2x = 6 \] Divide both sides by 2: \[ x = 3 \] Thus, the x-intercept is at (3, 0).

- Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y: \[ 2(0) + 3y = 6 \] Simplify to get: \[ 3y = 6 \] Divide both sides by 3: \[ y = 2 \] Thus, the y-intercept is at (0, 2).

- Plot the intercepts

Plot the x-intercept (3, 0) and the y-intercept (0, 2) on a Cartesian plane.

- Draw the line

Draw a straight line through the points (3, 0) and (0, 2) to represent the graph of the equation.

Key concepts

Intercepts

The intercepts of an equation are the points where the graph of the equation crosses the x-axis and y-axis. To find them, you set the opposite variable to zero.

For the x-intercept, set y = 0 and solve for x. Doing this for the equation 2x + 3y = 6, we get:

\[2x + 3(0) = 6\]

Simplify it to:

\[2x = 6\]

Divide by 2:

\[x = 3\]

So, the x-intercept is at (3, 0).

For the y-intercept, set x = 0 and solve for y. Doing this gives us:

\[2(0) + 3y = 6\]

Simplify it to:

\[3y = 6\]

Divide by 3:

\[y = 2\]

So, the y-intercept is at (0, 2).

These points are fundamental because they help you understand where the line will cross the axes.

Graphing

Graphing a linear equation involves plotting points and drawing a line through them. Here, we use the intercepts we found.

Start by drawing a Cartesian plane, which is simply a grid with a horizontal x-axis and a vertical y-axis.

Mark the x-intercept at (3, 0) and the y-intercept at (0, 2). These two points will guide your line.

Once you plot them, use a ruler to draw a straight line passing through these points. This line represents your linear equation 2x + 3y = 6. You can extend the line beyond these points as far as you like.

Graphing this way helps to visualize the solutions of the equation and understand how changes in variables affect the graph.

Cartesian Plane

The Cartesian plane, also called the coordinate plane, is the foundational tool for graphing equations. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).

The point where these axes intersect is called the origin, denoted by (0, 0).

When graphing, you place points according to their coordinates (x, y). The x-coordinate tells you how far to move right or left from the origin, and the y-coordinate tells you how far to move up or down.

For example, a point (3, 2) means moving 3 units to the right and 2 units up from the origin. Knowing how to navigate the Cartesian plane is key to successfully plotting points and understanding graphs.

Lines, curves, and other shapes can also be drawn on this plane, making it a versatile tool for all kinds of mathematical visualization.

Solving Equations

Solving linear equations means finding the values of variables that make the equation true. Here, we use simple algebraic steps.

For the given equation 2x + 3y = 6, we solved for x and y to find the intercepts. Here's a quick recap:

• Set y = 0 to solve for x.
• Set x = 0 to solve for y.

By isolating x and y in these steps, we found the intercepts (3, 0) and (0, 2).

To solve for x or y when the other variable is zero, perform basic operations like addition, subtraction, multiplication, and division.

Breaking down the problem step-by-step helps in understanding and solving more complex equations in the future. Being comfortable with these steps ensures better problem-solving skills in mathematics!