Problem 28 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$3 x-7 y=21$$

Step-by-Step Solution

Answer

The x-intercept is \((7, 0)\) and the y-intercept is \((0, -3)\).

Step 1 - Identify the equation format

The given equation is in the standard linear form: \(3x - 7y = 21\).

Step 2 - Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\).\[3x - 7(0) = 21\]\[3x = 21\]\[x = 7\]The x-intercept is \((7, 0)\).

Step 3 - Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\).\[3(0) - 7y = 21\]\[-7y = 21\]\[y = -3\]The y-intercept is \((0, -3)\).

Step 4 - Plot the intercepts on the graph

Plot the points \((7, 0)\) and \((0, -3)\) on the coordinate plane. These points are the intercepts.

Step 5 - Draw the line

Draw a straight line through the points \((7, 0)\) and \((0, -3)\). This line represents the equation \(3x - 7y = 21\).

Step 6 - Label the intercepts

Label the points \((7, 0)\) as the x-intercept and \((0, -3)\) as the y-intercept on the graph.

Key Concepts

X-Intercept

To understand the x-intercept, picture the point where the line crosses the x-axis on the coordinate plane. In simpler terms, it's where the line meets the horizontal line (x-axis). At this point, the value of y is always zero.

Finding the X-Intercept

• In any linear equation, set y = 0.
• Solve for x.

Taking our example equation, \(3x - 7y = 21\), we set y to zero and solve:

\[ 3x - 7(0) = 21 \] \[ 3x = 21 \]
\[ x = 7 \]
So, the x-intercept is \( (7, 0) \). This is the point where the line touches the x-axis.

Remember, on the graph, this is noted as (7, 0).

Y-Intercept

The y-intercept is where the line crosses the y-axis. This is the vertical line on the coordinate plane. At this point, the value of x is always zero.

Finding the Y-Intercept

• In any linear equation, set x = 0.
• Solve for y.

For our example equation \(3x - 7y = 21\), we set x to zero and solve:

\[ 3(0) - 7y = 21 \] \[ -7y = 21 \]
\[ y = -3 \]
So, the y-intercept is \( (0, -3) \). This is the point where the line touches the y-axis.

On the graph, we mark this as (0, -3).

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph points, lines, and curves. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).

Each point on the plane is defined by an ordered pair \( (x, y) \). The x value tells us how far to move horizontally, and the y value shows how far to move vertically.

Steps to Graph a Line

• Identify any intercepts, such as the x-intercept and y-intercept.
• Plot these intercepts on the coordinate plane.
• Draw a straight line through these points.

By plotting points such as \( (7, 0) \) and \( (0, -3) \), we extend an infinite line that represents our linear equation.

Standard Form of a Linear Equation

The standard form of a linear equation is represented as
\[Ax + By = C \]
Here, A, B, and C are constants. This format is handy for quickly finding intercepts and graphing lines.

Converting to Slope-Intercept Form
Sometimes it’s easier to understand and work with the equation when it is in slope-intercept form ( \( y = mx + b \) ), where m represents the slope and b represents the y-intercept.

• Our example equation \(3x - 7y = 21\) can be re-arranged as:
• \[-7y = -3x + 21 \]
• \[ y = \frac{3}{7}x - 3 \]

Both forms provide unique insights and are valuable tools for different mathematical tasks. While the standard form easily shows intercepts, the slope-intercept form reveals the rate of change and starting point.